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 |
| prime radical | #30628 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1604 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)(19,25)(20,21)(22,26)(23,27)(24,28)(29,32)(30,39)(31,41)(33,34)(35,42)(36,43)(37,44)(38,40)(45,48)(46,47);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19,24,31)(18,25,28,41)(20,32,27,40)(21,29,23,38)(22,35,37,45)(26,42,44,48)(30,34,46,36)(33,47,43,39);(17,20,21,28,24,27,23,18)(19,25,38,40,31,41,29,32)(22,33,34,44,37,43,36,26)(30,39,45,48,46,47,35,42);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,21,24,23)(18,20,28,27)(19,29,31,38)(22,36,37,34)(25,32,41,40)(26,43,44,33)(30,45,46,35)(39,48,47,42);(3,11)(6,9)(10,16)(14,15)(17,22,21,34,24,37,23,36)(18,26,20,33,28,44,27,43)(19,30,38,45,31,46,29,35)(25,39,40,48,41,47,32,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=fi,e^2=hi,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, 6716314);