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×#3285 | 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 | |
1008 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(15,23)(16,27)(17,30)(19,24)(20,28)(21,32)(25,34)(26,42)(29,33)(31,37)(35,41)(36,43)(38,40)(39,44);(6,9)(10,12)(13,14,18,22)(15,24,25,41)(16,28,31,44)(17,32,30,21)(19,34,35,23)(20,37,39,27)(26,43,33,38)(29,40,42,36);(2,4)(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,29)(17,33)(18,25)(20,38)(21,40)(22,35)(23,27)(24,28)(26,30)(31,42)(32,36)(34,37)(39,43)(41,44);(6,9)(10,12)(15,25)(19,35)(23,34)(24,41)(26,33)(29,42)(36,40)(38,43)Polycyclic presentation:
2^9|b^2=i,c^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=di,f^c=fh,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, 407328);