Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #513193 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #443 ◁ #2170 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,9,17)(3,18,24,74)(4,25,30,66)(5,31,38,99)(6,16,8,14)(7,41,44,67)(10,46,51,21)(11,39,55,28)(12,56,60,43)(13,61,65,36)(15,40,69,34)(19,75,26,48)(20,57,84,112)(22,87,45,73)(23,88,90,126)(27,58,93,122)(29,59,95,54)(32,68,76,127)(33,91,104,125)(35,98,37,97)(42,62,110,124)(47,82,52,117)(49,86,70,101)(50,107,116,100)(53,108,118,102)(63,106,64,105)(71,80,114,78)(72,77,111,89)(79,123,85,115)(81,119,83,120)(92,96,103,121)(94,113,109,128);(1,3)(2,10)(4,26)(5,32)(6,22)(7,42)(8,45)(9,24)(11,52)(12,57)(13,62)(14,49)(15,68)(16,70)(17,51)(18,31)(19,30)(20,27)(21,36)(23,34)(25,72)(28,80)(29,94)(33,89)(35,100)(37,107)(38,76)(39,78)(40,90)(41,50)(43,85)(44,110)(46,61)(47,55)(48,58)(53,115)(54,119)(56,79)(59,120)(60,112)(63,88)(64,126)(65,124)(66,111)(67,116)(69,127)(71,121)(73,97)(74,99)(75,122)(77,104)(81,92)(82,102)(83,103)(84,93)(86,106)(87,98)(91,113)(95,109)(96,114)(101,105)(108,117)(118,123)(125,128);(1,4)(2,11)(3,19)(5,27)(6,39)(7,29)(8,28)(9,30)(10,47)(12,41)(13,53)(14,66)(15,54)(16,25)(17,55)(18,71)(20,76)(21,77)(22,80)(23,79)(24,26)(31,96)(32,84)(33,36)(34,43)(35,103)(37,92)(38,93)(40,56)(42,109)(44,95)(45,78)(46,89)(48,73)(49,72)(50,112)(51,52)(57,116)(58,98)(59,69)(60,67)(61,104)(62,123)(63,125)(64,91)(65,118)(68,120)(70,111)(74,114)(75,87)(81,100)(82,86)(83,107)(85,90)(88,113)(94,110)(97,122)(99,121)(101,117)(102,105)(106,108)(115,124)(119,127)(126,128);(1,4)(2,12)(3,20)(5,33)(6,28)(7,43)(8,39)(9,30)(10,48)(11,41)(13,53)(14,59)(15,25)(16,54)(17,60)(18,72)(19,76)(21,85)(22,81)(23,89)(24,84)(26,32)(27,36)(29,34)(31,58)(35,102)(37,108)(38,104)(40,95)(42,82)(44,56)(45,83)(46,79)(47,73)(49,71)(50,115)(51,75)(52,87)(55,67)(57,62)(61,93)(63,91)(64,125)(65,118)(66,69)(68,113)(70,114)(74,111)(77,90)(78,107)(80,100)(86,109)(88,120)(92,106)(94,101)(96,98)(97,121)(99,122)(103,105)(110,117)(112,124)(116,123)(119,126)(127,128)Polycyclic presentation:
2^9|a^2=i,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fi,g^a=gh,h^a=hi,c^b=ci,d^b=de,e^b=eg,g^b=gh,h^b=hi,f^c=fi,e^d=eg,g^d=gh,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420390);