Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437418 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,10)(7,11,12,9)(13,14)(15,24)(16,21)(17,22)(18,31)(19,23)(20,26)(25,38)(27,34)(28,36)(29,37)(30,41)(32,35)(33,39)(40,43)(42,44);(1,2)(3,4)(13,15,19,27)(14,20,23,35)(16,25,29,40)(17,26,18,32)(21,33,37,44)(22,34,31,24)(28,39,30,42)(36,43,41,38);(2,4)(5,7,8,12)(6,9,10,11)(13,16)(14,21)(15,25)(17,28)(18,30)(19,29)(20,33)(22,36)(23,37)(24,38)(26,39)(27,40)(31,41)(32,42)(34,43)(35,44);(2,4)(13,17,19,18)(14,22,23,31)(15,26,27,32)(16,28,29,30)(20,34,35,24)(21,36,37,41)(25,39,40,42)(33,43,44,38);(16,29)(21,37)(25,40)(28,30)(33,44)(36,41)(38,43)(39,42)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6352575);