Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #394 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | C4×#394 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1331 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,29)(3,14,17,9,19,32,18,33)(4,20,25,36,24,48,23,10)(5,11,27,37,28,38,26,53)(15,30,42,55,21,34,43,39)(16,40,44,31,46,56,45,57)(22,47,52,59,51,63,50,35)(41,54,61,64,49,58,62,60);(1,3)(2,9)(4,21)(5,16)(6,18)(7,17)(8,19)(10,34)(11,31)(12,32)(13,33)(14,29)(15,24)(20,39)(22,49)(23,42)(25,43)(26,45)(27,44)(28,46)(30,36)(35,58)(37,56)(38,57)(40,53)(41,51)(47,60)(48,55)(50,61)(52,62)(54,59)(63,64);(1,4,6,23,8,24,7,25)(2,10,12,20,13,36,29,48)(3,15,17,42,19,21,18,43)(5,22,26,50,28,51,27,52)(9,30,14,39,33,34,32,55)(11,35,37,47,38,59,53,63)(16,41,44,61,46,49,45,62)(31,54,40,60,57,58,56,64);(1,5,7,27,8,28,6,26)(2,11,12,37,13,38,29,53)(3,16,17,44,19,46,18,45)(4,22,25,52,24,51,23,50)(9,31,32,56,33,57,14,40)(10,35,20,47,36,59,48,63)(15,41,42,61,21,49,43,62)(30,54,55,64,34,58,39,60)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266357);