Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667676 |
| 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 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(3,4)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,25)(21,24)(22,26)(27,32)(28,31)(29,30);(1,2)(3,7)(4,8)(5,6)(10,13)(11,15)(17,19,22,27)(18,23,26,32)(20,29,21,28)(24,31,25,30);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(18,24)(19,27)(20,21)(23,30)(25,26)(31,32);(1,4,5,3)(2,8,6,7)(10,13)(14,16);(2,6)(7,8)(17,20,22,21)(18,25,26,24)(19,28,27,29)(23,31,32,30)Polycyclic presentation:
2^9|b^2=i,c^2=f,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6679883);