Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1987 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,14,18,22)(15,19,26,36)(16,20,30,40)(17,21,32,42)(23,33,43,44)(24,34,25,35)(27,37,28,38)(29,39,31,41);(6,9)(10,12)(13,15)(14,19)(16,27)(17,24)(18,26)(20,37)(21,34)(22,36)(23,29)(25,32)(28,30)(31,43)(33,39)(35,42)(38,40)(41,44);(2,4)(13,16,18,30)(14,20,22,40)(15,23,26,43)(17,31,32,29)(19,33,36,44)(21,41,42,39)(24,27,25,28)(34,37,35,38);(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,28)(17,24)(18,26)(20,38)(21,34)(22,36)(23,31)(25,32)(27,30)(29,43)(33,41)(35,42)(37,40)(39,44);(15,24)(17,32)(19,34)(21,42)(23,27)(25,26)(28,43)(29,31)(33,37)(35,36)(38,44)(39,41)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482998);