Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10,13,11)(12,15,16,14)(17,18,22,26)(19,27,20,30)(21,32,28,25)(23,31,24,29);(1,3)(2,5)(4,7)(6,8)(9,11,13,10)(12,14,16,15)(17,19,28,31,22,20,21,29)(18,23,32,30,26,24,25,27);(9,12,13,16)(10,14,11,15);(3,7)(5,8)(10,11)(14,15)(19,20)(23,24)(27,30)(29,31);(17,20,21,31,22,19,28,29)(18,24,25,30,26,23,32,27)Polycyclic presentation:
2^9|a^2=h,b^2=fgi,c^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7497946);