Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,17,21,18,22,29,36)(15,19,25,32,23,30,37,41)(16,20,28,35,27,34,40,44)(24,31,38,42,26,33,39,43);(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,26)(17,25)(18,23)(20,33)(21,32)(22,30)(24,27)(28,39)(29,37)(31,34)(35,43)(36,41)(38,40)(42,44);(2,4)(15,23)(19,30)(24,26)(25,37)(31,33)(32,41)(38,39)(42,43);(7,12)(9,11)(13,16)(14,20)(15,24)(17,28)(18,27)(19,31)(21,35)(22,34)(23,26)(25,38)(29,40)(30,33)(32,42)(36,44)(37,39)(41,43);(6,10)(9,11)(15,23)(16,27)(19,30)(20,34)(25,37)(28,40)(32,41)(35,44)Polycyclic presentation:
2^9|a^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=dfi,e^b=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6293164);