Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417601 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2651 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6)(7,19)(8,24)(9,15)(10,17)(11,16)(12,18)(13,22)(14,26)(20,34)(21,47)(23,44)(25,55)(27,45)(28,60)(29,43)(30,42)(31,41)(32,36)(33,39)(35,52)(37,51)(38,58)(40,53)(46,50)(48,61)(49,54)(56,63)(57,65)(59,64)(62,67)(66,68);(5,7,10,22,12,23,11,32)(6,13,16,19,18,36,17,44)(8,20,26,49,27,51,33,50)(9,28,29,53,30,52,31,21)(14,37,24,54,39,34,45,46)(15,40,41,60,42,47,43,35)(25,56,57,67,59,68,58,48)(38,63,64,61,65,66,55,62);(5,8)(6,14)(7,20)(9,25)(10,26)(11,33)(12,27)(13,34)(15,38)(16,24)(17,45)(18,39)(19,46)(21,48)(22,49)(23,51)(28,56)(29,57)(30,59)(31,58)(32,50)(35,61)(36,37)(40,66)(41,64)(42,65)(43,55)(44,54)(47,63)(52,68)(53,67)(60,62);(2,4)(5,9,12,30)(6,15,18,42)(7,21,23,53)(8,25,27,59)(10,31,11,29)(13,35,36,60)(14,38,39,65)(16,43,17,41)(19,47,44,40)(20,48,51,67)(22,52,32,28)(24,55,45,64)(26,58,33,57)(34,61,37,62)(46,63,54,66)(49,68,50,56)Polycyclic presentation:
2^9|b^2=e,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,f^c=fi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325052);