Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632074 |
| 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 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1461 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)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18,22,27)(19,23,28,37)(20,25,32,41)(21,24,34,26)(29,36,31,40)(30,38,43,35)(33,39,46,42)(44,48,45,47);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,31)(21,30)(22,28)(24,38)(25,40)(26,35)(27,37)(29,32)(33,45)(34,43)(36,41)(39,47)(42,48)(44,46);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(18,24)(19,28)(21,34)(23,35)(25,39)(26,27)(29,31)(33,46)(36,47)(37,38)(40,48)(41,42);(2,5)(6,8)(10,13)(14,16)(17,20,22,32)(18,25,27,41)(19,29,28,31)(21,33,34,46)(23,36,37,40)(24,39,26,42)(30,44,43,45)(35,47,38,48);(9,11,12,15)(10,14,13,16)(20,32)(25,41)(29,31)(33,46)(36,40)(39,42)(44,45)(47,48)Polycyclic presentation:
2^9|a^2=i,c^2=f,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^b=dfh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6560965);