Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14572 |
| prime radical | #23438 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,18,9,20,33,19,34)(4,21,24,10,26,37,25,38)(5,11,29,40,30,42,28,41)(16,44,46,59,47,31,22,35)(17,43,27,56,48,39,49,32)(23,52,53,36,55,50,54,61)(45,58,51,62,64,63,57,60);(1,3)(2,9)(4,22)(5,27)(6,19)(7,18)(8,20)(10,35)(11,39)(12,33)(13,15)(14,34)(16,24)(17,28)(21,31)(23,45)(25,47)(26,46)(29,48)(30,49)(32,40)(36,58)(37,44)(38,59)(41,56)(42,43)(50,62)(51,53)(52,60)(54,57)(55,64)(61,63);(1,4,7,25,8,26,6,24)(2,10,12,21,14,38,13,37)(3,16,18,22,20,47,19,46)(5,23,28,53,30,55,29,54)(9,31,33,59,34,44,15,35)(11,36,41,50,42,61,40,52)(17,45,49,51,48,64,27,57)(32,58,39,62,56,63,43,60);(1,5)(2,11)(3,17)(4,23)(6,29)(7,28)(8,30)(9,32)(10,36)(12,41)(13,40)(14,42)(15,43)(16,45)(18,49)(19,27)(20,48)(21,50)(22,51)(24,54)(25,53)(26,55)(31,58)(33,39)(34,56)(35,60)(37,52)(38,61)(44,63)(46,57)(47,64)(59,62)Polycyclic presentation:
2^9|a^2=h,c^2=egh,e^2=i,f^2=i,g^2=i,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,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297622);