Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| 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 | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2026 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)(15,24)(16,19)(17,29)(18,30)(20,23)(21,27)(22,32)(25,34)(26,35)(28,37)(31,36)(33,43)(38,41)(39,42)(40,44);(6,9)(7,11)(13,15,19,27)(14,21,16,24)(17,26,32,41)(18,28,20,34)(22,35,29,38)(23,37,30,25)(31,42,33,44)(36,40,43,39);(2,4)(13,16,19,14)(15,25,27,37)(17,22,32,29)(18,30,20,23)(21,34,24,28)(26,40,41,39)(31,36,33,43)(35,42,38,44);(6,9)(10,12)(13,17,19,32)(14,22,16,29)(15,26,27,41)(18,31,20,33)(21,35,24,38)(23,36,30,43)(25,39,37,40)(28,42,34,44);(5,7,8,11)(6,10,9,12)(13,18)(14,23)(15,27)(16,30)(17,31)(19,20)(22,36)(25,37)(26,41)(29,43)(32,33)(39,40)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6579995);