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 | |
567 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)(7,10)(8,9)(11,12)(13,14,17,20)(15,24,25,36)(16,21,31,38)(18,32,30,23)(19,34,35,26)(22,40,39,29)(27,44,33,41)(28,42,43,37);(1,2)(3,4)(13,15,17,25)(14,19,20,35)(16,28,31,43)(18,33,30,27)(21,24,38,36)(22,41,39,44)(23,42,32,37)(26,40,34,29);(2,4)(6,9)(10,12)(14,20)(19,35)(21,38)(22,39)(23,32)(24,36)(37,42)(41,44);(5,7,8,11)(6,10,9,12)(14,21)(15,25)(16,29)(18,30)(19,36)(20,38)(22,23)(24,35)(26,43)(28,34)(31,40)(32,39)(37,41)(42,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=fi,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, 411569);