Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#532 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1768 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)(3,7)(4,6)(5,8)(9,10,13,14,12,15,11,16)(17,18)(19,25)(20,24)(21,23)(22,26)(27,31)(28,30)(29,32);(3,8)(5,7)(17,19)(18,22)(20,27)(21,26)(23,25)(24,29)(28,32)(30,31);(1,3,4,8)(2,5,6,7)(9,11,12,13)(10,14,15,16)(19,26)(22,25)(27,32)(29,31);(2,6)(5,7)(10,15)(14,16)(18,23)(19,26)(24,30)(27,32);(10,15)(14,16)(17,20,21,28)(18,24,23,30)(19,27,26,32)(22,29,25,31)Polycyclic presentation:
2^9|a^2=h,c^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=cfgi,d^b=df104 conjugacy classes — view the full character table.
| maximal subgroup of | #72311750, #72311809, #72313409, … |
| maximal normal subgroup of | #72311750, #72311809, #72313409, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314853);