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 | C22×D8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
642 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)(7,9)(8,10)(11,12)(13,14,16,19)(15,24,25,37)(17,20,28,34)(18,32,29,23)(21,26,35,43)(22,39,36,30)(27,40,33,42)(31,38,44,41);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(14,19)(20,34)(21,35)(22,36)(23,32)(24,37)(38,41)(40,42);(2,4)(5,8,9,12)(6,10,7,11)(14,20)(17,30)(18,29)(19,34)(21,37)(22,23)(24,35)(26,31)(27,33)(28,39)(32,36)(38,42)(40,41)(43,44);(6,7)(10,11)(13,15,16,25)(14,21,19,35)(17,31,28,44)(18,33,29,27)(20,37,34,24)(22,40,36,42)(23,41,32,38)(26,39,43,30)Polycyclic presentation:
2^9|a^2=i,b^2=ei,c^2=ei,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=dei,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, 416695);