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)(15,25)(16,22)(17,24)(18,20)(19,23)(21,27)(26,39)(28,38)(29,37)(30,34)(31,36)(32,33)(35,42)(40,43)(41,44);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(14,20)(21,33)(22,34)(23,24)(25,38)(35,43)(36,37)(39,44);(2,4)(13,15,18,28)(14,21,20,33)(16,26,30,41)(17,27,19,32)(22,35,34,43)(23,25,24,38)(29,40,31,42)(36,44,37,39);(2,4)(13,16)(14,22)(15,26)(17,31)(18,30)(19,29)(20,34)(21,35)(23,37)(24,36)(25,39)(27,42)(28,41)(32,40)(33,43)(38,44);(13,17,18,19)(14,23,20,24)(15,27,28,32)(16,29,30,31)(21,25,33,38)(22,36,34,37)(26,40,41,42)(35,44,43,39)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,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, 6345326);