Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234345 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1589 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10)(11,15)(12,16)(13,14)(17,18,24,29)(19,25,30,20)(21,27,26,32)(22,28,23,31);(1,2)(3,7)(4,6)(5,8)(10,14)(11,12)(17,19,24,30)(18,25,29,20)(21,22,26,23)(27,28,32,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,16)(12,15)(13,14)(17,20,23,27,24,25,22,32)(18,26,28,19,29,21,31,30);(3,8)(5,7)(9,11,13,12)(10,15,14,16)(17,21)(18,27)(19,22)(20,31)(23,30)(24,26)(25,28)(29,32);(2,6)(5,7)(9,12,13,11)(10,15,14,16)(17,22,24,23)(18,28,29,31)(19,21,30,26)(20,32,25,27)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7161987);