Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1802 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)(11,14)(12,13)(15,16)(17,18,21,25,20,24,28,32)(19,23,27,31,26,22,29,30);(1,3)(2,5)(4,8)(6,7)(17,19)(18,22)(20,26)(21,27)(23,24)(25,30)(28,29)(31,32);(3,8)(5,7)(10,13)(14,16)(17,19)(18,23)(20,26)(21,27)(22,24)(25,31)(28,29)(30,32);(2,6)(5,7)(18,24)(19,26)(25,32)(27,29);(9,11,12,15)(10,14,13,16)(18,24)(22,23)(25,32)(30,31)Polycyclic presentation:
2^9|a^2=h,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=cfi,d^b=di,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72308682, #72308886 |
| maximal normal subgroup of | #72308682, #72308886 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6282453);