Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #381470 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14299 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1334 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)(3,15,18,9,20,33,19,32)(4,21,24,10,25,36,23,37)(5,11,28,39,29,40,27,41)(16,30,45,42,47,34,46,58)(17,43,49,31,50,60,51,61)(22,52,53,35,54,38,26,55)(44,57,48,63,64,62,56,59);(1,3)(2,9)(4,16)(5,17)(6,19)(7,18)(8,20)(10,34)(11,31)(12,33)(13,15)(14,32)(21,42)(22,44)(23,46)(24,45)(25,47)(26,56)(27,51)(28,49)(29,50)(30,37)(35,62)(36,58)(38,59)(39,60)(40,61)(41,43)(48,53)(52,63)(54,64)(55,57);(1,4,8,25)(2,10,14,37)(3,16,20,47)(5,26,29,53)(6,24,7,23)(9,30,32,34)(11,38,40,52)(12,21,13,36)(15,42,33,58)(17,48,50,56)(18,46,19,45)(22,28,54,27)(31,59,61,63)(35,41,55,39)(43,57,60,62)(44,51,64,49);(1,5,7,28,8,29,6,27)(2,11,12,39,14,40,13,41)(3,17,18,49,20,50,19,51)(4,22,24,53,25,54,23,26)(9,31,33,60,32,61,15,43)(10,35,36,38,37,55,21,52)(16,44,45,48,47,64,46,56)(30,57,42,63,34,62,58,59)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,e^b=ei,d^c=de,e^c=ei,f^c=fi,h^c=hi,f^d=fi,g^d=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288669);