Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,10)(7,11,8,12)(13,14)(15,24)(16,27)(17,32)(18,22)(19,35)(20,26)(21,31)(23,34)(25,41)(28,38)(29,36)(30,44)(33,40)(37,43)(39,42);(1,2)(3,4)(6,10)(11,12)(13,15,18,26)(14,20,22,24)(16,28,30,29)(17,19,33,34)(21,37,39,41)(23,40,35,32)(25,42,43,31)(27,36,44,38);(5,7,9,8)(6,11,10,12)(13,16,19,31,18,30,34,42)(14,21,23,27,22,39,35,44)(15,25,17,28,26,43,33,29)(20,36,32,37,24,38,40,41);(2,4)(13,17,19,26,18,33,34,15)(14,20,23,32,22,24,35,40)(16,29,31,25,30,28,42,43)(21,38,27,41,39,36,44,37);(5,8,9,7)(6,11,10,12)(14,22)(16,30)(20,24)(23,35)(25,43)(28,29)(31,42)(32,40)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fgi,d^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472083);