Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2072 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)(13,14,17,22)(15,24,27,19)(16,29,30,21)(18,33,32,23)(20,34,38,31)(25,39,26,36)(28,37,44,41)(35,42,40,43);(1,2)(3,4)(6,9)(7,11)(13,15,17,27)(14,19,22,24)(16,25,30,26)(18,28,32,44)(20,35,38,40)(21,39,29,36)(23,37,33,41)(31,42,34,43);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,25)(17,30)(18,34)(19,35)(21,23)(22,38)(24,40)(26,27)(28,43)(29,33)(31,32)(36,41)(37,39)(42,44);(6,9)(10,12)(13,16)(14,21)(15,26)(17,30)(18,31)(19,36)(20,23)(22,29)(24,39)(25,27)(28,43)(32,34)(33,38)(35,41)(37,40)(42,44);(2,4)(15,27)(16,30)(19,24)(20,38)(21,29)(28,44)(31,34)(37,41)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^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,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6576596);