Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628851 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,24)(15,20,28,36)(16,29,30,37)(17,23,18,22)(21,32,38,31)(25,39,40,43)(26,35,27,34)(33,42,44,41);(6,9)(10,12)(13,15,19,28)(14,20,24,36)(16,25,30,40)(17,26,18,27)(21,33,38,44)(22,34,23,35)(29,39,37,43)(31,41,32,42);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,25)(17,32)(18,31)(19,30)(20,33)(22,37)(23,29)(24,38)(26,42)(27,41)(28,40)(34,43)(35,39)(36,44);(14,22)(17,18)(20,34)(21,37)(23,24)(26,27)(29,38)(31,32)(33,43)(35,36)(39,44)(41,42);(16,30)(21,38)(25,40)(29,37)(31,32)(33,44)(39,43)(41,42)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,h^a=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536434);