Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | #23920 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1792 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,8,4)(17,18,22,24)(19,21,26,29)(20,23,27,30)(25,28,31,32);(2,6)(4,7)(9,10)(11,15)(12,14)(13,16);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,24,22,26,29,18)(20,25,28,30,27,31,32,23);(2,6)(3,8)(11,16)(13,15)(17,20,22,27)(18,23,24,30)(19,25,26,31)(21,28,29,32);(1,4,5,7)(2,3,6,8)(10,14)(13,15)(20,27)(23,30)(25,31)(28,32)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72308665, #72308931, #72309026 |
| maximal normal subgroup of | #72308665, #72308931, #72309026 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6819543);