Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14391 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,28)(3,9,17,32,18,34,16,33)(4,19,23,10,22,38,29,37)(5,24,26,40,27,39,25,11)(14,41,44,30,43,35,20,50)(15,45,47,58,48,59,46,31)(21,36,54,56,53,62,55,49)(42,57,52,60,63,64,51,61);(1,3)(2,9)(4,20)(5,15)(6,16)(7,17)(8,18)(10,35)(11,31)(12,32)(13,34)(14,23)(19,30)(21,52)(22,44)(24,45)(25,46)(26,47)(27,48)(28,33)(29,43)(36,61)(37,41)(38,50)(39,59)(40,58)(42,55)(49,64)(51,53)(54,63)(56,57)(60,62);(1,4)(2,10)(3,14)(5,21)(6,29)(7,23)(8,22)(9,30)(11,36)(12,38)(13,37)(15,42)(16,20)(17,44)(18,43)(19,28)(24,56)(25,55)(26,54)(27,53)(31,57)(32,35)(33,41)(34,50)(39,49)(40,62)(45,60)(46,51)(47,52)(48,63)(58,64)(59,61);(1,5,6,25,8,27,7,26)(2,11,12,24,13,40,28,39)(3,15,16,46,18,48,17,47)(4,21,29,55,22,53,23,54)(9,31,32,45,34,58,33,59)(10,36,38,56,37,62,19,49)(14,42,20,51,43,63,44,52)(30,57,35,60,50,64,41,61)Polycyclic presentation:
2^9|a^2=h,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292032);