Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6660385 |
| 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 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1770 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,19)(20,27,28,32)(21,26,31,24)(23,30,25,29);(2,5)(6,7)(11,15)(13,16)(17,19,22,18)(20,23,28,25)(21,26,31,24)(27,30,32,29);(3,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,20,22,28)(18,23,19,25)(21,30,31,29)(24,32,26,27);(1,3,4,8)(2,6,5,7)(11,15)(13,16)(20,28)(23,25)(27,32)(29,30);(2,5)(6,7)(10,14)(13,16)(18,19)(23,25)(24,26)(27,32)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,e^b=ei,d^c=dfh,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6567923);