Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6807031 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1619 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)(3,9,11,6)(4,12,13,7)(10,14,16,15)(17,18)(19,24)(20,32)(21,25)(22,26)(23,27)(28,38)(29,43)(30,33)(31,39)(34,46)(35,40)(36,41)(37,42)(44,47)(45,48);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,24)(20,33)(21,35)(22,29)(23,31)(25,40)(26,43)(27,39)(28,37)(30,32)(34,47)(36,45)(38,42)(41,48)(44,46);(17,20,22,18,23,26,32,27)(19,24,30,33,31,39,29,43)(21,34,37,25,36,42,46,41)(28,38,35,40,44,47,45,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(21,36)(25,41)(28,44)(34,42)(35,45)(37,46)(38,47)(40,48);(3,11)(6,9)(10,16)(14,15)(17,21,22,37,23,36,32,46)(18,25,26,42,27,41,20,34)(19,28,30,35,31,44,29,45)(24,38,33,40,39,47,43,48)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=f,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6716299);