Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499724 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
677 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 44 points:
(5,6,8,10)(7,11,12,9)(13,14,19,24)(15,25,28,44)(16,29,31,41)(17,32,35,43)(18,36,26,23)(20,33,38,27)(21,39,37,30)(22,40,42,34);(1,2)(3,4)(13,15,19,28)(14,20,24,38)(16,18,31,26)(17,30,35,39)(21,36,37,23)(22,41,42,29)(25,43,44,32)(27,40,33,34);(2,4)(5,7,8,12)(6,9,10,11);(13,16,19,31)(14,21,24,37)(15,26,28,18)(17,33,35,27)(20,23,38,36)(22,25,42,44)(29,43,41,32)(30,40,39,34)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=f,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,e^b=eh,h^b=hi,e^d=eh,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410839);