Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996405 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,10,14)(4,7,12,16)(9,13,11,15)(17,18,22,29,24,30,23,28)(19,31,33,25,35,47,34,43)(20,36,38,26,40,48,39,45)(21,27,32,42,41,46,37,44);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19)(18,25)(20,37)(21,39)(22,34)(23,33)(24,35)(26,44)(27,36)(28,47)(29,31)(30,43)(32,40)(38,41)(42,45)(46,48);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,20)(18,26)(19,32)(21,34)(22,38)(23,39)(24,40)(25,42)(27,31)(28,36)(29,48)(30,45)(33,41)(35,37)(43,44)(46,47);(4,12)(7,16)(9,11)(13,15)(17,21,23,37,24,41,22,32)(18,27,28,44,30,46,29,42)(19,20,34,39,35,40,33,38)(25,26,31,36,43,45,47,48);(2,8)(3,10)(7,16)(9,11)(17,22,24,23)(18,28,30,29)(19,33,35,34)(20,38,40,39)(21,32,41,37)(25,31,43,47)(26,36,45,48)(27,44,46,42)Polycyclic presentation:
2^9|a^2=fh,d^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6941833);