Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #510223 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31396 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #575 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
481 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,8,16,9,13,6,14)(3,10,20,45,19,47,41,60)(4,22,26,51,25,61,42,11)(5,28,30,12,32,53,33,52)(7,35,39,55,40,15,34,59)(17,62,65,43,66,104,64,103)(18,67,70,44,29,88,68,106)(21,72,36,96,71,46,97,56)(23,77,73,109,80,75,79,48)(24,49,84,76,81,113,82,111)(27,86,37,50,85,57,99,101)(31,92,91,58,38,89,95,54)(63,120,121,119,90,102,123,124)(69,115,100,107,74,117,93,105)(78,108,126,128,122,110,125,127)(83,114,94,118,87,112,98,116);(1,3)(2,10,13,47)(4,23)(5,29,32,18)(6,20)(7,36)(8,41)(9,19)(11,48,51,109)(12,44)(14,45,16,60)(15,56,35,96)(17,27)(21,39)(22,75,61,77)(24,78,81,122)(25,80)(26,79)(28,88)(30,70,33,68)(31,93,38,100)(34,71)(37,64)(40,97)(42,73)(43,101,103,50)(46,55,72,59)(49,110)(52,106)(53,67)(54,115)(57,62,86,104)(58,117)(63,83,90,87)(65,99)(66,85)(69,91,74,95)(76,128)(82,126,84,125)(89,107)(92,105)(94,121,98,123)(102,112)(108,113)(111,127)(114,120)(116,119)(118,124);(1,4,6,26,9,25,8,42)(2,11,16,61,13,51,14,22)(3,17,20,64,19,66,41,65)(5,24,33,84,32,81,30,82)(7,37,34,85,40,99,39,27)(10,43,60,104,47,103,45,62)(12,49,53,111,52,113,28,76)(15,57,59,50,35,86,55,101)(18,63,70,123,29,90,68,121)(21,73,36,23,71,79,97,80)(31,94,95,87,38,98,91,83)(44,102,67,124,106,120,88,119)(46,77,96,109,72,75,56,48)(54,116,92,112,58,118,89,114)(69,125,100,122,74,126,93,78)(105,127,117,108,107,128,115,110);(1,5)(2,12)(3,18)(4,24)(6,33)(7,38)(8,30)(9,32)(10,44)(11,49)(13,52)(14,28)(15,58)(16,53)(17,63)(19,29)(20,70)(21,74)(22,76)(23,78)(25,81)(26,84)(27,87)(31,40)(34,91)(35,54)(36,93)(37,98)(39,95)(41,68)(42,82)(43,102)(45,88)(46,107)(47,106)(48,108)(50,114)(51,113)(55,92)(56,117)(57,118)(59,89)(60,67)(61,111)(62,119)(64,123)(65,121)(66,90)(69,71)(72,105)(73,126)(75,127)(77,128)(79,125)(80,122)(83,85)(86,116)(94,99)(96,115)(97,100)(101,112)(103,120)(104,124)(109,110)Polycyclic presentation:
2^9|a^2=h,b^2=efh,c^2=ehi,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gh,h^b=hi,e^c=ei,g^c=gh,h^c=hi,f^d=fi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417421);