Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996671 |
| 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 |
| prime radical | C2×#127 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1508 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)(7,8)(9,10,14,11)(12,15,13,16)(17,18,22,26)(19,27,28,32)(20,25,21,24)(23,29,31,30);(1,2,5,6)(3,7,4,8)(10,11)(12,13)(19,28)(23,31)(27,32)(29,30);(1,3,5,4)(2,7,6,8)(9,11,14,10)(12,15,13,16)(17,19,21,30,22,28,20,29)(18,23,25,27,26,31,24,32);(1,4,5,3)(2,7,6,8)(9,12,14,13)(10,15,11,16);(9,13,14,12)(10,15,11,16)(17,20,22,21)(18,24,26,25)(19,29,28,30)(23,32,31,27)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=fg,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7101467);