Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 | C2×#3203 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1869 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)(7,10)(8,9)(11,12)(13,14,17,19)(15,23,24,39)(16,28,29,21)(18,32,25,22)(20,34,33,27)(26,41,30,38)(31,37,40,43)(35,42,36,44);(1,2)(3,4)(6,9)(7,11)(14,19)(15,24)(16,29)(22,32)(27,34)(31,40)(35,36)(38,41);(2,4)(13,15,25,34,17,24,18,27)(14,20,32,39,19,33,22,23)(16,30,40,42,29,26,31,44)(21,36,43,38,28,35,37,41);(6,9)(10,12)(13,16)(14,21)(15,26)(17,29)(18,31)(19,28)(20,35)(22,37)(23,38)(24,30)(25,40)(27,42)(32,43)(33,36)(34,44)(39,41);(2,4)(5,7,8,11)(6,10,9,12)(15,24)(16,29)(20,33)(21,28)(23,39)(27,34)(31,40)(37,43)Polycyclic presentation:
2^9|a^2=i,c^2=gi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592681);