Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13873 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1899 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,9,7,11,5,12)(3,13,16,24,14,8,20,22)(4,17,18,25,19,10,21,26)(15,23,30,32,28,27,29,31);(1,3,5,16,7,14,6,20)(2,8,12,22,11,13,9,24)(4,15,18,29,19,28,21,30)(10,23,26,31,17,27,25,32);(2,9)(3,14)(5,6)(8,22)(10,25)(11,12)(13,24)(15,28)(17,26)(18,21)(23,31)(27,32);(1,4)(2,10)(3,15)(5,18)(6,21)(7,19)(8,23)(9,25)(11,17)(12,26)(13,27)(14,28)(16,29)(20,30)(22,31)(24,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,h^b=hi,f^c=fi,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72319119, #72319399, #72319471, … |
| maximal normal subgroup of | #72319119, #72319399, #72319471, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267486);