Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500689 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#447 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,15,16,13)(17,18,23,28)(19,24,32,39)(20,25,34,41)(21,26,22,27)(29,36,31,38)(30,37,45,48)(33,40,35,42)(43,46,44,47);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,32)(18,24,28,39)(20,30,34,45)(21,31,22,29)(25,37,41,48)(26,38,27,36)(33,43,35,44)(40,46,42,47);(10,14)(13,15)(19,29)(21,22)(24,36)(26,27)(30,43)(31,32)(33,35)(37,46)(38,39)(40,42)(44,45)(47,48);(2,5)(6,8)(11,16)(13,15)(17,20)(18,25)(19,30)(21,35)(22,33)(23,34)(24,37)(26,42)(27,40)(28,41)(29,43)(31,44)(32,45)(36,46)(38,47)(39,48);(1,3,4,7)(2,6,5,8)(17,21,23,22)(18,26,28,27)(19,31,32,29)(20,33,34,35)(24,38,39,36)(25,40,41,42)(30,43,45,44)(37,46,48,47)Polycyclic presentation:
2^9|a^2=fi,b^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7419261);