Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | |
1791 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,23,29)(19,20,31,25)(21,27,26,32)(22,30,24,28);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,31)(18,25,29,20)(21,24,26,22)(27,30,32,28);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,20,22,32,23,25,24,27)(18,26,28,19,29,21,30,31);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,21,23,26)(18,27,29,32)(19,22,31,24)(20,30,25,28);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,22,23,24)(18,28,29,30)(19,26,31,21)(20,27,25,32)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6939187);