Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1189 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,14,17)(11,18,21,15)(12,16,13,22)(19,24,20,23)(25,26,29,31)(27,35,36,49)(28,40,41,33)(30,43,37,34)(32,45,44,39)(38,52,50,48)(42,47,51,55)(46,53,54,56);(1,2,4,5)(3,7,8,6)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(26,31)(27,36)(33,40)(34,43)(38,50)(39,45)(47,55)(53,56);(2,5)(6,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,27,37,45,29,36,30,39)(26,32,43,49,31,44,34,35)(28,38,51,56,41,50,42,53)(33,46,55,52,40,54,47,48);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,28)(26,33)(27,38)(29,41)(30,42)(31,40)(32,46)(34,47)(35,48)(36,50)(37,51)(39,53)(43,55)(44,54)(45,56)(49,52);(1,3,4,8)(2,6,5,7)(11,21)(15,18)(19,20)(23,24)(28,41)(33,40)(38,50)(42,51)(46,54)(47,55)(48,52)(53,56)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=fg,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6712118);