Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#127 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,7)(3,6,4,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26)(27,35)(28,33)(29,31)(30,34)(32,37)(36,45)(38,49)(39,53)(40,43)(41,48)(42,44)(46,52)(47,50)(51,55)(54,56);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(26,31)(27,36)(28,39)(30,42)(32,45)(33,48)(35,47)(37,50)(40,41)(43,53)(46,49)(51,54);(1,4,5,3)(2,6,7,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,27,30,36,29,37,42,50)(26,32,34,47,31,35,44,45)(28,38,41,51,40,52,39,54)(33,46,48,56,43,49,53,55);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(26,31)(27,37)(33,43)(34,44)(36,50)(38,52)(48,53)(51,54);(11,21)(15,18)(19,20)(23,24)(25,28,30,41,29,40,42,39)(26,33,34,48,31,43,44,53)(27,38,36,51,37,52,50,54)(32,46,47,56,35,49,45,55)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=fgh,d^2=fi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6727668);