Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1721 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)(3,7)(4,6)(5,8)(19,25)(20,27)(22,29)(24,30)(26,31)(28,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18,21,23)(19,26,28,22)(20,24,27,30)(25,31,32,29);(2,6)(3,8)(10,13)(14,16)(17,19,27,32,21,28,20,25)(18,22,30,31,23,26,24,29);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,23)(19,28)(24,30)(25,32);(10,13)(14,16)(18,23)(22,26)(24,30)(29,31)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319809 |
| maximal normal subgroup of | #72319809 |
| maximal quotient of | #72319983 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6693582);