Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499724 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
611 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12);(6,9)(10,11)(13,14,19,23)(15,24,28,42)(16,21,18,34)(17,30,33,41)(20,35,37,26)(22,29,40,32)(25,39,27,44)(31,38,43,36);(2,4)(13,15,19,28)(14,20,23,37)(16,27,18,25)(17,31,33,43)(21,38,34,36)(22,39,40,44)(24,41,42,30)(26,29,35,32);(5,7,8,12)(6,10,9,11)(13,16,19,18)(14,21,23,34)(15,25,28,27)(17,29,33,32)(20,36,37,38)(22,30,40,41)(24,39,42,44)(26,43,35,31)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408686);