Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10,13,15)(11,14,12,16)(17,18,23,28)(19,24,32,41)(20,25,22,27)(21,34,36,26)(29,38,30,39)(31,46,47,40)(33,44,37,42)(35,48,45,43);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,30)(21,35)(22,29)(23,32)(25,39)(26,43)(27,38)(28,41)(31,33)(34,48)(36,45)(37,47)(40,42)(44,46);(9,12,13,11)(10,14,15,16)(19,29)(20,22)(24,38)(25,27)(30,32)(31,35)(33,37)(39,41)(40,43)(42,44)(45,47)(46,48);(1,3,4,7)(2,6,5,8)(17,20,23,22)(18,25,28,27)(19,30,32,29)(21,33,36,37)(24,39,41,38)(26,42,34,44)(31,45,47,35)(40,48,46,43);(2,5)(6,8)(17,21,23,36)(18,26,28,34)(19,31,32,47)(20,33,22,37)(24,40,41,46)(25,42,27,44)(29,35,30,45)(38,43,39,48)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=fi,e^2=i,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, 6600347);