Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394813 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2436 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14719 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1555 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,15)(4,21)(5,26)(6,14)(7,12)(8,13)(9,18)(10,23)(11,27)(16,30)(17,46)(19,33)(20,32)(22,52)(24,36)(25,37)(28,39)(29,38)(31,49)(34,45)(35,54)(40,44)(41,47)(42,60)(43,50)(48,57)(51,53)(55,59)(56,61)(58,62)(63,64);(1,3,6,18,8,20,7,19)(2,9,12,15,13,33,14,32)(4,16,23,43,25,45,24,44)(5,17,27,47,28,49,29,48)(10,34,21,50,36,30,37,40)(11,31,26,41,38,46,39,57)(22,42,53,61,54,63,55,62)(35,58,51,64,52,56,59,60);(1,4,6,23,8,25,7,24)(2,10,12,21,13,36,14,37)(3,16,18,43,20,45,19,44)(5,22,27,53,28,54,29,55)(9,30,15,40,33,34,32,50)(11,35,26,51,38,52,39,59)(17,42,47,61,49,63,48,62)(31,56,41,60,46,58,57,64);(1,5)(2,11)(3,17)(4,22)(6,27)(7,29)(8,28)(9,31)(10,35)(12,26)(13,38)(14,39)(15,41)(16,42)(18,47)(19,48)(20,49)(21,51)(23,53)(24,55)(25,54)(30,56)(32,57)(33,46)(34,58)(36,52)(37,59)(40,60)(43,61)(44,62)(45,63)(50,64)Polycyclic presentation:
2^9|b^2=e,c^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302010);