Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509065 |
| 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 | #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 | |
725 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)(13,14,18,23)(15,24,27,42)(16,19,31,35)(17,32,30,22)(20,36,38,25)(21,39,34,28)(26,43,33,40)(29,41,44,37);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11);(2,4)(14,19)(16,28)(17,30)(20,24)(21,22)(23,35)(25,29)(26,33)(31,39)(32,34)(36,44)(37,43)(38,42)(40,41);(6,10)(7,12)(13,15,18,27)(14,20,23,38)(16,29,31,44)(17,33,30,26)(19,24,35,42)(21,40,34,43)(22,41,32,37)(25,39,36,28)Polycyclic presentation:
2^9|a^2=i,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=de,f^c=fh,h^c=hi,f^d=fh,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416270);