Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1872 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)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,23,31,39)(20,32,36,42)(21,25,34,26)(24,40,43,35)(28,45,33,48)(29,38,30,44)(37,46,41,47);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,31)(18,23,27,39)(20,33,36,28)(21,30,34,29)(24,41,43,37)(25,44,26,38)(32,48,42,45)(35,46,40,47);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,20,34,40,22,36,21,35)(18,24,25,42,27,43,26,32)(19,28,29,46,31,33,30,47)(23,37,44,48,39,41,38,45);(1,4,5,3)(2,8,6,7)(11,15)(13,16)(17,19)(18,23)(20,28)(21,30)(22,31)(24,37)(25,44)(26,38)(27,39)(29,34)(32,45)(33,36)(35,47)(40,46)(41,43)(42,48);(17,21,22,34)(18,25,27,26)(19,29,31,30)(20,35,36,40)(23,38,39,44)(24,42,43,32)(28,46,33,47)(37,45,41,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6367551);