Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574024 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1704 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,28)(20,33)(21,26)(22,38)(23,27)(24,32)(25,35)(29,45)(30,44)(31,40)(34,43)(36,48)(37,42)(39,46)(41,47);(9,11)(10,13)(12,16)(14,15)(17,19,21,31,23,30,22,32)(18,24,38,44,27,40,26,28)(20,34,36,47,35,29,37,46)(25,41,48,43,33,39,42,45);(2,5)(6,8)(11,16)(13,14)(17,20)(18,25)(19,29)(21,36)(22,37)(23,35)(24,39)(26,42)(27,33)(28,43)(30,34)(31,46)(32,47)(38,48)(40,41)(44,45);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,16)(13,15)(19,30)(20,35)(24,40)(25,33)(28,44)(31,32)(36,37)(42,48);(11,16)(13,14)(17,21,23,22)(18,26,27,38)(19,31,30,32)(20,36,35,37)(24,28,40,44)(25,42,33,48)(29,46,34,47)(39,43,41,45)Polycyclic presentation:
2^9|b^2=fh,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^d=ehi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6519667);