Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806575 |
| 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 | |
1310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10,12,13)(11,14,15,16)(17,18,22,26)(19,25,20,24)(21,31,30,27)(23,28,29,32);(1,2)(3,6)(4,5)(7,8)(17,19,22,20)(18,24,26,25)(21,29,30,23)(27,32,31,28);(10,13)(14,16)(17,20,23,21,22,19,29,30)(18,25,28,27,26,24,32,31);(1,3,4,7)(2,6,5,8)(18,26)(24,25)(27,31)(28,32);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21,29,20,22,30,23,19)(18,27,32,25,26,31,28,24)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=fi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6740251);