Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384166 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2377 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14453 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
761 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)(3,13,18,8)(4,19)(5,10,24,35)(6,11)(7,26)(9,22)(12,14)(15,38,42,28)(16,29)(17,30,27,37)(20,31,49,45)(21,46,50,32)(23,33)(25,52,40,36)(34,44)(39,47)(41,54)(43,60)(48,57)(51,58,61,63)(53,55)(56,64)(59,62);(1,3,14,17,6,18,7,27)(2,8,12,37,11,13,26,30)(4,20,39,42,22,49,23,15)(5,16,40,43,24,44,25,53)(9,31,33,28,19,45,47,38)(10,34,36,55,35,29,52,60)(21,48,61,62,50,64,51,41)(32,57,58,59,46,56,63,54);(1,4,7,23,6,22,14,39)(2,9,26,47,11,19,12,33)(3,15,17,20,18,42,27,49)(5,21,25,51,24,50,40,61)(8,28,37,45,13,38,30,31)(10,32,52,63,35,46,36,58)(16,41,43,48,44,62,53,64)(29,54,60,57,34,59,55,56);(1,5)(2,10)(3,16)(4,21)(6,24)(7,25)(8,29)(9,32)(11,35)(12,36)(13,34)(14,40)(15,41)(17,43)(18,44)(19,46)(20,48)(22,50)(23,51)(26,52)(27,53)(28,54)(30,55)(31,56)(33,58)(37,60)(38,59)(39,61)(42,62)(45,57)(47,63)(49,64)Polycyclic presentation:
2^9|a^2=h,b^2=efg,c^2=eg,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294385);