Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682596 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1243 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)(25,26,29,33)(27,35,37,31)(28,39,41,54)(30,44,43,34)(32,47,49,42)(36,51,40,55)(38,46,53,50)(45,52,48,56);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,29,37)(26,31,33,35)(28,40,41,36)(30,38,43,53)(32,48,49,45)(34,46,44,50)(39,51,54,55)(42,56,47,52);(2,6)(7,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,30,42,29,41,43,47)(26,32,34,39,33,49,44,54)(27,36,38,52,37,40,53,56)(31,45,46,55,35,48,50,51);(1,3,5,4)(2,7,6,8)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(26,33)(27,37)(28,41)(34,44)(38,53)(42,47)(45,48)(51,55);(1,4,5,3)(2,7,6,8)(11,21)(15,18)(19,20)(23,24)(28,41)(32,49)(36,40)(39,54)(42,47)(45,48)(51,55)(52,56)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6787282);