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)(25,26,30,31)(27,36,37,48)(28,40,42,33)(29,34,44,35)(32,46,45,39)(38,53,52,51)(41,50,43,49)(47,55,56,54);(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)(27,37)(33,40)(34,35)(38,52)(39,46)(49,50)(54,55);(2,5)(6,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,27,29,39,30,37,44,46)(26,32,35,36,31,45,34,48)(28,38,43,55,42,52,41,54)(33,47,49,51,40,56,50,53);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,28)(26,33)(27,38)(29,43)(30,42)(31,40)(32,47)(34,50)(35,49)(36,51)(37,52)(39,55)(41,44)(45,56)(46,54)(48,53);(1,3,4,8)(2,6,5,7)(11,21)(15,18)(19,20)(23,24)(25,29,30,44)(26,34,31,35)(27,39,37,46)(28,41,42,43)(32,48,45,36)(33,49,40,50)(38,54,52,55)(47,51,56,53)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=fgi,d^2=fi,e^2=h,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, 6712121);