Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1230 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)(25,26,30,34)(27,35,39,51)(28,32,41,48)(29,43,36,33)(31,44,46,38)(37,50,40,54)(42,56,52,49)(45,55,47,53);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24);(2,6)(7,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,27,36,44,30,39,29,38)(26,31,43,51,34,46,33,35)(28,40,52,53,41,37,42,55)(32,47,56,50,48,45,49,54);(1,3,5,4)(2,7,6,8)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,28)(26,32)(27,37)(29,42)(30,41)(31,45)(33,49)(34,48)(35,50)(36,52)(38,53)(39,40)(43,56)(44,55)(46,47)(51,54);(1,4,5,3)(2,7,6,8)(11,21)(15,18)(19,20)(23,24)(28,41)(32,48)(37,40)(42,52)(45,47)(49,56)(50,54)(53,55)Polycyclic presentation:
2^9|a^2=fi,c^2=fgi,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6787001);