Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32950 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2847 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14680 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3339 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)(3,15,19,32,20,31,18,9)(4,21,23,10,25,35,24,34)(5,11,27,36,28,37,26,38)(16,41,43,39,45,29,44,55)(17,40,47,56,48,57,46,30)(22,50,51,58,52,49,53,33)(42,54,61,64,62,60,63,59);(1,3,6,18,8,20,7,19)(2,9,12,15,14,32,13,31)(4,16,23,43,25,45,24,44)(5,17,26,46,28,48,27,47)(10,29,21,39,34,41,35,55)(11,30,36,40,37,56,38,57)(22,42,51,61,52,62,53,63)(33,54,49,59,58,60,50,64);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,25)(9,29)(11,33)(12,21)(13,35)(14,34)(15,39)(17,42)(18,43)(19,44)(20,45)(26,51)(27,53)(28,52)(30,54)(31,55)(32,41)(36,49)(37,58)(38,50)(40,59)(46,61)(47,63)(48,62)(56,60)(57,64);(1,5)(2,11)(3,17)(4,22)(6,26)(7,27)(8,28)(9,30)(10,33)(12,36)(13,38)(14,37)(15,40)(16,42)(18,46)(19,47)(20,48)(21,49)(23,51)(24,53)(25,52)(29,54)(31,57)(32,56)(34,58)(35,50)(39,59)(41,60)(43,61)(44,63)(45,62)(55,64)Polycyclic presentation:
2^9|a^2=h,b^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72227317, #72280993, #72281021, … |
| maximal normal subgroup of | #72280993, #72281021, #72281077, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299396);