Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431707 |
| 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 | |
2862 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)(15,20)(16,21)(17,22)(18,24)(19,23)(25,33)(26,34)(27,36)(28,35)(29,37)(30,38)(31,39)(32,40)(41,43)(42,44);(6,9)(10,12)(13,15,19,28)(14,20,23,35)(16,29,31,42)(17,27,18,26)(21,37,39,44)(22,36,24,34)(25,32,41,30)(33,40,43,38);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,25)(17,30)(18,32)(19,31)(20,33)(22,38)(23,39)(24,40)(26,29)(27,42)(28,41)(34,37)(35,43)(36,44);(13,17,19,18)(14,22,23,24)(15,26,28,27)(16,30,31,32)(20,34,35,36)(21,38,39,40)(25,29,41,42)(33,37,43,44);(14,23)(20,35)(21,39)(22,24)(33,43)(34,36)(37,44)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,h^b=hi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339366);