Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1621 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(17,18,22,20)(19,26,27,32)(21,24,28,25)(23,31,30,29);(1,3,7,4)(2,5,8,6)(9,10)(11,15)(12,14)(13,16)(19,27)(23,30)(26,32)(29,31);(1,4,7,3)(2,5,8,6)(9,11)(10,13)(12,16)(14,15)(17,19,28,31,22,27,21,29)(18,23,24,32,20,30,25,26);(1,5,7,6)(2,3,8,4)(11,16)(13,15)(17,20,22,18)(19,23,27,30)(21,24,28,25)(26,31,32,29);(1,6,7,5)(2,3,8,4)(10,14)(13,15)(17,21,22,28)(18,24,20,25)(19,29,27,31)(23,32,30,26)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fgi,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6822972);