Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31282 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2684 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14427 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
993 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,14,6,12)(3,15,19,9,20,33,18,34)(4,21,24,37,25,42,30,10)(5,11,27,39,29,41,28,40)(16,31,46,43,22,35,47,60)(17,38,26,57,48,44,49,32)(23,52,56,36,55,50,54,63)(45,59,53,62,51,61,58,64);(1,3)(2,9)(4,22)(5,26)(6,19)(7,18)(8,20)(10,35)(11,38)(12,33)(13,15)(14,34)(16,25)(17,27)(21,43)(23,53)(24,46)(28,48)(29,49)(30,47)(31,37)(32,39)(36,62)(40,57)(41,44)(42,60)(45,56)(50,59)(51,54)(52,61)(55,58)(63,64);(1,4)(2,10)(3,16,20,22)(5,23)(6,24)(7,30)(8,25)(9,31,34,35)(11,36)(12,21)(13,42)(14,37)(15,43,33,60)(17,45,48,51)(18,46,19,47)(26,58,49,53)(27,54)(28,56)(29,55)(32,59,57,61)(38,64,44,62)(39,52)(40,50)(41,63);(1,5)(2,11)(3,17)(4,23)(6,28)(7,27)(8,29)(9,32)(10,36)(12,40)(13,39)(14,41)(15,44)(16,45)(18,49)(19,26)(20,48)(21,50)(22,51)(24,56)(25,55)(30,54)(31,59)(33,38)(34,57)(35,61)(37,63)(42,52)(43,62)(46,53)(47,58)(60,64)Polycyclic presentation:
2^9|a^2=h,c^2=efg,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=ei,h^b=hi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293562);