Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2749 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14591 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
755 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)(3,8,16,29)(4,9)(5,10,23,35)(6,11)(7,25)(12,13)(14,27,41,55)(15,42)(17,44,38,30)(18,46,48,31)(19,32,49,61)(20,34)(21,51)(22,28)(24,37,26,36)(33,39)(40,63)(43,56)(45,57)(47,58)(50,62,52,60)(53,54)(59,64);(1,3,13,38,6,16,7,17)(2,8,25,44,11,29,12,30)(4,18,21,41,20,48,39,14)(5,22,26,43,23,15,24,45)(9,31,33,55,34,46,51,27)(10,28,37,57,35,42,36,56)(19,47,50,40,49,64,52,53)(32,59,60,63,61,58,62,54);(1,4,7,21,6,20,13,39)(2,9,12,33,11,34,25,51)(3,14,38,48,16,41,17,18)(5,19,26,52,23,49,24,50)(8,27,44,46,29,55,30,31)(10,32,37,62,35,61,36,60)(15,40,43,64,22,53,45,47)(28,54,56,59,42,63,57,58);(1,5)(2,10)(3,15)(4,19)(6,23)(7,26)(8,28)(9,32)(11,35)(12,37)(13,24)(14,40)(16,22)(17,45)(18,47)(20,49)(21,52)(25,36)(27,54)(29,42)(30,57)(31,58)(33,62)(34,61)(38,43)(39,50)(41,53)(44,56)(46,59)(48,64)(51,60)(55,63)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=gh,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^c=ei,f^c=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298146);