Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573511 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1971 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14,19,24)(15,20,27,36)(16,21,31,40)(17,22,26,35)(18,23,32,41)(25,34,29,38)(28,37,43,44)(30,39,33,42);(5,7,8,12)(6,9,10,11)(13,15,19,27)(14,20,24,36)(16,28,31,43)(17,32,26,18)(21,37,40,44)(22,41,35,23)(25,33,29,30)(34,42,38,39);(2,4)(13,16,19,31)(14,21,24,40)(15,25,27,29)(17,28,26,43)(18,33,32,30)(20,34,36,38)(22,37,35,44)(23,42,41,39);(13,17,19,26)(14,22,24,35)(15,18,27,32)(16,29,31,25)(20,23,36,41)(21,38,40,34)(28,33,43,30)(37,42,44,39);(15,26)(17,27)(18,32)(20,35)(22,36)(23,41)(25,43)(28,29)(30,33)(34,44)(37,38)(39,42)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=i,d^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481697);