Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470077 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1802 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18,20,25,21,24,30,23)(19,26,28,32,27,31,29,22);(2,5)(6,8)(10,14)(11,12)(17,19)(18,22)(20,29)(21,27)(23,26)(24,32)(25,31)(28,30);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(18,23)(19,27)(20,30)(22,31)(24,25)(26,32);(9,12,13,11)(10,16,14,15)(19,27)(22,32)(26,31)(28,29);(10,14)(15,16)(18,24)(22,32)(23,25)(26,31)Polycyclic presentation:
2^9|a^2=g,c^2=fgi,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6391722);