Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,10)(7,11,12,9)(13,14,18,24)(15,22,27,20)(16,28,30,38)(17,23,19,32)(21,33,39,29)(25,40,42,44)(26,36,31,35)(34,43,37,41);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,25)(17,31)(18,27)(19,26)(21,34)(22,24)(23,36)(28,44)(29,43)(30,42)(32,35)(33,41)(37,39)(38,40);(2,4)(13,16,18,30)(14,21,24,39)(15,25,27,42)(17,29,19,33)(20,34,22,37)(23,38,32,28)(26,41,31,43)(35,44,36,40);(2,4)(13,15)(14,22)(16,25)(17,31)(18,27)(19,26)(20,24)(21,37)(23,35)(28,40)(29,43)(30,42)(32,36)(33,41)(34,39)(38,44);(13,17,18,19)(14,23,24,32)(15,26,27,31)(16,29,30,33)(20,35,22,36)(21,38,39,28)(25,41,42,43)(34,44,37,40)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345334);