Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456670 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,10,17,12,16,11,32)(7,13,21,36,20,38,31,37)(8,22,25,40,27,44,26,14)(9,28,29,15,30,43,33,42)(18,45,47,61,23,54,53,34)(19,48,51,62,50,63,52,35)(24,39,56,55,57,41,58,59)(46,64,66,68,49,60,67,65);(1,2)(3,4)(5,7,11,21,12,20,10,31)(6,13,17,37,16,38,32,36)(8,23,26,53,27,18,25,47)(9,19,33,51,30,50,29,52)(14,34,22,54,40,61,44,45)(15,35,43,63,42,62,28,48)(24,49,58,67,57,46,56,66)(39,60,55,68,41,64,59,65);(2,4)(5,8,12,27)(6,14,16,40)(7,18,20,23)(9,24,30,57)(10,26,11,25)(13,34,38,61)(15,41,42,39)(17,44,32,22)(19,49,50,46)(21,53,31,47)(28,59,43,55)(29,58,33,56)(35,60,62,64)(36,54,37,45)(48,68,63,65)(51,66,52,67);(5,9)(6,15)(7,19)(8,24)(10,29)(11,33)(12,30)(13,35)(14,39)(16,42)(17,43)(18,46)(20,50)(21,51)(22,55)(23,49)(25,56)(26,58)(27,57)(28,32)(31,52)(34,60)(36,48)(37,63)(38,62)(40,41)(44,59)(45,65)(47,66)(53,67)(54,68)(61,64)Polycyclic presentation:
2^9|a^2=f,b^2=efi,c^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,f^c=fi,h^c=hi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363869);