Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #433970 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15494 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,5,10,6,18)(3,8,19,23,13,22,14,27)(4,9,16,28,17,25,20,26)(12,21,15,31,29,32,30,24);(1,3)(2,8)(4,15)(5,13)(6,19)(7,14)(9,24)(10,22)(11,27)(12,16)(17,30)(18,23)(20,29)(21,26)(25,31)(28,32);(1,4)(2,9,10,25)(3,12)(5,17)(6,20)(7,16)(8,21,22,32)(11,28,18,26)(13,29)(14,30)(15,19)(23,31,27,24);(2,10)(3,13)(6,7)(8,22)(9,25)(12,29)(16,20)(21,32)Polycyclic presentation:
2^9|a^2=eg,c^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72286431, #72287261, #72287354, … |
| maximal normal subgroup of | #72286431, #72287261, #72287354, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341166);