Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | #36018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,19,23)(15,20,26,35)(16,27,18,21)(17,22,30,38)(24,32,31,33)(25,34,41,44)(28,37,29,36)(39,42,40,43);(5,7,8,12)(6,9,10,11)(15,24)(16,18)(17,28)(20,32)(21,27)(22,37)(26,31)(29,30)(33,35)(36,38)(39,40)(42,43);(1,2,3,4)(13,15,19,26)(14,20,23,35)(16,24,18,31)(17,25,30,41)(21,33,27,32)(22,34,38,44)(28,39,29,40)(36,43,37,42);(6,10)(9,11)(13,16,19,18)(14,21,23,27)(15,24,26,31)(17,29,30,28)(20,33,35,32)(22,37,38,36)(25,40,41,39)(34,42,44,43);(7,12)(9,11)(13,17,19,30)(14,22,23,38)(15,25,26,41)(16,28,18,29)(20,34,35,44)(21,36,27,37)(24,39,31,40)(32,42,33,43)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590599);