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 | C22×D8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
777 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)(5,6,9,12)(7,11,8,10)(13,14,18,22)(15,23,25,42)(16,27,31,41)(17,32,30,21)(19,34,35,24)(20,37,39,28)(26,43,33,38)(29,40,44,36);(5,7,9,8)(6,10,12,11)(15,24)(16,28)(17,30)(19,23)(20,27)(21,32)(25,34)(26,44)(29,33)(31,37)(35,42)(36,43)(38,40)(39,41);(2,4)(13,15,18,25)(14,19,22,35)(16,29,31,44)(17,33,30,26)(20,38,39,43)(21,40,32,36)(23,41,42,27)(24,37,34,28);(5,8,9,7)(6,11,12,10)(15,25)(19,35)(23,42)(24,34)(26,33)(29,44)(36,40)(38,43)Polycyclic presentation:
2^9|a^2=ei,b^2=e,c^2=i,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=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, 408109);