Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499724 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3341 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
781 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)(5,6,8,9)(7,11,12,10)(13,14,18,22)(15,23,26,37)(16,28,30,44)(17,31,29,21)(19,32,33,24)(20,35,36,25)(27,42,40,39)(34,41,38,43);(6,9)(10,11)(15,24)(16,25)(17,29)(19,23)(20,28)(21,31)(26,32)(27,43)(30,35)(33,37)(34,42)(36,44)(38,39)(40,41);(1,2,3,4)(13,15,25,41,17,27,30,32,18,26,35,43,29,40,16,24)(14,19,28,39,21,34,36,37,22,33,44,42,31,38,20,23);(5,7,8,12)(6,10,9,11)(15,26)(19,33)(23,37)(24,32)(27,40)(34,38)(39,42)(41,43)Polycyclic presentation:
2^9|a^2=ei,c^2=fh,d^2=e,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, 410272);