Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2218 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)(17,30)(22,36)(27,41)(29,32)(31,37)(34,43)(39,44)(40,42);(5,7)(6,9)(8,12)(10,11)(13,14,19,24)(15,25,28,38)(16,23,18,21)(17,31,30,37)(20,33,35,26)(22,32,36,29)(27,34,41,43)(39,40,44,42);(1,2,3,4)(13,15,19,28)(14,20,24,35)(16,26,18,33)(17,27,30,41)(21,25,23,38)(22,34,36,43)(29,40,32,42)(31,39,37,44);(6,10)(9,11)(13,16,19,18)(14,21,24,23)(15,26,28,33)(17,32,30,29)(20,25,35,38)(22,37,36,31)(27,42,41,40)(34,44,43,39);(7,12)(9,11)(13,17,19,30)(14,22,24,36)(15,27,28,41)(16,29,18,32)(20,34,35,43)(21,31,23,37)(25,39,38,44)(26,40,33,42)Polycyclic presentation:
2^9|b^2=i,c^2=gi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589535);