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 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26,30,34)(27,35,38,31)(28,39,42,46)(29,43,44,33)(32,40,48,36)(37,45,51,49)(41,54,55,52)(47,50,56,53);(3,4)(6,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,30,38)(26,31,34,35)(28,40,42,36)(29,37,44,51)(32,46,48,39)(33,45,43,49)(41,53,55,50)(47,54,56,52);(1,3,5,4)(2,6,8,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,27,36,30,42,38,40)(26,32,31,39,34,48,35,46)(29,41,37,50,44,55,51,53)(33,47,45,52,43,56,49,54);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,29)(26,33)(27,37)(28,41)(30,44)(31,45)(32,47)(34,43)(35,49)(36,50)(38,51)(39,52)(40,53)(42,55)(46,54)(48,56);(1,4,5,3)(2,7,8,6)(11,21)(15,18)(19,20)(23,24)(29,44)(33,43)(37,51)(41,55)(45,49)(47,56)(50,53)(52,54)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fgh,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, 6757056);