Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1881 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,25)(21,26)(22,27)(23,29)(28,32)(30,31);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(19,28)(20,22)(23,31)(24,32)(25,27)(29,30);(10,14)(13,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,26)(25,31)(27,32);(1,4,5,3)(2,6,7,8)(17,19)(18,24)(20,28)(21,29)(22,30)(23,26)(25,32)(27,31);(17,20,21,22)(18,25,26,27)(19,28,29,30)(23,31,24,32)Polycyclic presentation:
2^9|b^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348548);