Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,15)(11,14,12,16)(17,18)(19,30)(20,26)(21,37)(22,39)(23,29)(24,27)(25,33)(28,31)(32,42)(34,48)(35,47)(36,46)(38,44)(40,45)(41,43);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19)(18,25)(20,34)(21,32)(22,40)(23,33)(24,41)(26,44)(27,42)(28,45)(29,30)(31,35)(36,38)(37,43)(39,47)(46,48);(9,12,13,11)(10,14,15,16)(17,20)(18,26)(19,31)(21,35)(22,33)(23,36)(24,40)(25,39)(27,45)(28,30)(29,46)(32,38)(34,41)(37,47)(42,44)(43,48);(2,6)(5,7)(17,21,23,24)(18,27,29,37)(19,32,33,41)(20,35,36,40)(22,34,31,38)(25,42,30,43)(26,45,46,47)(28,48,39,44);(3,8)(5,7)(17,22,23,31)(18,28,29,39)(19,20,33,36)(21,38,24,34)(25,26,30,46)(27,44,37,48)(32,40,41,35)(42,47,43,45)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,d^2=i,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,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6599764);