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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2015 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)(7,11)(8,12)(9,10)(13,14,19,24)(15,20,27,36)(16,21,30,39)(17,22,32,41)(18,23,26,35)(25,34,43,44)(28,37,31,40)(29,38,33,42);(6,10)(11,12)(13,15,19,27)(14,20,24,36)(16,28,30,31)(17,18,32,26)(21,37,39,40)(22,23,41,35)(25,33,43,29)(34,42,44,38);(2,4)(13,16,19,30)(14,21,24,39)(15,25,27,43)(17,31,32,28)(18,33,26,29)(20,34,36,44)(22,40,41,37)(23,42,35,38);(5,7,9,8)(6,11,10,12)(13,17,19,32)(14,22,24,41)(15,26,27,18)(16,25,30,43)(20,35,36,23)(21,34,39,44)(28,29,31,33)(37,38,40,42);(5,8,9,7)(6,12,10,11)(15,17)(18,26)(20,22)(23,35)(25,31)(27,32)(28,43)(29,33)(34,40)(36,41)(37,44)(38,42)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=fi,e^2=f,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, 6485025);