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)(16,20,31,38)(17,32,30,22)(19,34,36,25)(21,39,37,29)(26,43,33,40)(28,41,44,35);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11);(2,4)(13,15,18,27)(14,19,23,36)(16,28,31,44)(17,33,30,26)(20,24,38,42)(21,40,37,43)(22,41,32,35)(25,39,34,29);(6,10)(7,12)(14,20)(16,29)(17,30)(19,24)(21,22)(23,38)(25,28)(26,33)(31,39)(32,37)(34,44)(35,43)(36,42)(40,41)Polycyclic presentation:
2^9|a^2=i,c^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, 416271);