Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509081 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #24266 | 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 | |
722 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)(7,11)(8,10)(9,12)(13,14,19,33,31,41,29,20,18,24,16,22,17,23,34,37)(15,25,43,42,32,40,30,39,28,36,26,38,27,44,35,21);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11);(2,4)(14,20)(16,29)(17,31)(19,34)(21,36)(22,23)(24,37)(25,39)(26,30)(27,32)(33,41)(35,43)(38,44)(40,42);(6,10)(7,12)(13,15,18,28)(14,21,24,39)(16,30,19,35)(17,32,31,27)(20,36,37,25)(22,40,33,44)(23,42,41,38)(26,34,43,29)Polycyclic presentation:
2^9|a^2=fi,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417008);