Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509119 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #30400 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #24296 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,9,16)(3,17,22,47)(4,24,29,82)(5,30,35,60)(6,15,8,13)(7,37,41,58)(10,23,46,20)(11,48,52,28)(12,39,57,100)(14,36,61,40)(18,66,25,83)(19,70,72,94)(21,45,77,109)(26,86,90,116)(27,91,42,59)(31,97,73,118)(32,55,34,96)(33,54,101,56)(38,62,104,51)(43,103,49,75)(44,107,108,71)(50,92,115,88)(53,99,64,76)(63,85,110,69)(65,74,102,78)(67,105,84,112)(68,106,79,80)(81,111,124,93)(87,117,98,113)(89,95,128,114)(119,127,125,121)(120,122,126,123);(1,3,8,23,9,22,6,20)(2,10,15,47,16,46,13,17)(4,25,27,79,29,18,42,68)(5,31,34,19,35,73,32,72)(7,21,40,78,41,77,36,74)(11,49,51,110,52,43,62,63)(12,53,56,44,57,64,54,108)(14,45,37,102,61,109,58,65)(24,80,59,83,82,106,91,66)(26,87,88,67,90,98,92,84)(28,85,38,103,48,69,104,75)(30,94,55,118,60,70,96,97)(33,99,39,71,101,76,100,107)(50,113,86,105,115,117,116,112)(81,119,95,122,124,125,114,123)(89,127,93,120,128,121,111,126);(1,4)(2,11)(3,18)(5,26)(6,27)(7,38)(8,42)(9,29)(10,43)(12,50)(13,51)(14,59)(15,62)(16,52)(17,63)(19,67)(20,68)(21,75)(22,25)(23,79)(24,37)(28,40)(30,95)(31,98)(32,88)(33,93)(34,92)(35,90)(36,48)(39,89)(41,104)(44,105)(45,66)(46,49)(47,110)(53,117)(54,86)(55,81)(56,116)(57,115)(58,82)(60,114)(61,91)(64,113)(65,80)(69,78)(70,122)(71,121)(72,84)(73,87)(74,85)(76,120)(77,103)(83,109)(94,123)(96,124)(97,119)(99,126)(100,128)(101,111)(102,106)(107,127)(108,112)(118,125);(1,5,9,35)(2,12,16,57)(3,19,22,72)(4,26,29,90)(6,34,8,32)(7,39,41,100)(10,44,46,108)(11,50,52,115)(13,56,15,54)(14,60,61,30)(17,64,47,53)(18,67,25,84)(20,73,23,31)(21,76,77,99)(24,81,82,124)(27,92,42,88)(28,93,48,111)(33,36,101,40)(37,55,58,96)(38,89,104,128)(43,105,49,112)(45,97,109,118)(51,116,62,86)(59,114,91,95)(63,113,110,117)(65,94,102,70)(66,119,83,125)(68,87,79,98)(69,121,85,127)(71,74,107,78)(75,120,103,126)(80,123,106,122)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=i,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fh,h^c=hi,e^d=ei,f^d=fh,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416318);