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 | |
1731 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)(17,18,22,28,24,29,23,40)(19,30,32,45,33,44,35,25)(20,34,36,26,37,46,41,43)(21,27,38,42,39,47,31,48);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19,24,33)(18,25,29,45)(20,31,37,38)(21,36,39,41)(22,35,23,32)(26,42,43,48)(27,34,47,46)(28,44,40,30);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,20)(18,26)(19,31)(21,32)(22,36)(23,41)(24,37)(25,42)(27,44)(28,46)(29,43)(30,47)(33,38)(34,40)(35,39)(45,48);(4,12)(7,16)(9,11)(13,15)(17,21,23,31,24,39,22,38)(18,27,40,48,29,47,28,42)(19,20,35,41,33,37,32,36)(25,43,44,46,45,26,30,34);(2,8)(3,10)(7,16)(9,11)Polycyclic presentation:
2^9|a^2=fh,b^2=i,d^2=fi,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, 6941832);