Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499724 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
793 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 44 points:
(2,4)(5,6,9,12)(7,11,8,10)(13,14)(15,23)(16,19)(17,30)(18,32)(20,24)(21,31)(22,29)(25,34)(26,35)(27,44)(28,36)(33,40)(37,43)(38,42)(39,41);(5,7,9,8)(6,10,12,11)(14,19)(15,24)(17,31)(18,29)(20,35)(21,36)(23,33)(25,28)(26,40)(27,43)(30,34)(37,39)(38,44)(41,42);(1,2,3,4)(5,7,9,8)(6,11,12,10)(13,15,25,43,18,27,17,24,16,26,31,42,29,41,28,40)(14,20,36,44,22,37,21,23,19,33,30,39,32,38,34,35);(5,8,9,7)(6,11,12,10)(14,19)(15,26)(21,34)(22,32)(24,40)(27,41)(30,36)(42,43)Polycyclic presentation:
2^9|a^2=ei,b^2=ei,c^2=efh,d^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410746);