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)(15,20,28,37)(16,21,32,41)(17,22,27,36)(18,23,26,35)(25,34,30,39)(29,38,43,44)(31,40,33,42);(6,10)(11,12)(13,15)(14,20)(16,29)(17,18)(19,28)(21,38)(22,23)(24,37)(25,31)(26,27)(30,33)(32,43)(34,40)(35,36)(39,42)(41,44);(2,4)(13,16,19,32)(14,21,24,41)(15,25,28,30)(17,29,27,43)(18,33,26,31)(20,34,37,39)(22,38,36,44)(23,42,35,40);(5,7,9,8)(6,11,10,12)(13,17)(14,22)(15,26)(16,30)(18,28)(19,27)(20,35)(21,39)(23,37)(24,36)(25,32)(29,31)(33,43)(34,41)(38,40)(42,44);(5,8,9,7)(6,12,10,11)(15,27)(17,28)(18,26)(20,36)(22,37)(23,35)(25,43)(29,30)(31,33)(34,44)(38,39)(40,42)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=f,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, 6485018);