Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458577 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16320 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1018 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)(3,8,14,23)(4,9)(5,12)(6,10)(7,11)(13,29,30,21)(15,22,19,24)(16,25,31,32)(17,26)(18,27)(20,28);(1,3)(2,8,11,23)(4,16)(5,19)(6,15)(7,14)(9,25,27,32)(10,22,12,24)(13,17)(18,31)(20,30)(21,26,29,28);(1,4)(2,9)(3,13,14,30)(5,17)(6,20)(7,18)(8,21,23,29)(10,28)(11,27)(12,26)(15,16,19,31)(22,32,24,25);(2,10)(5,6)(8,22)(9,26)(11,12)(13,30)(15,19)(17,20)(21,32)(23,24)(25,29)(27,28)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237073, #72237078, #72237090, … |
| maximal normal subgroup of | #72237073, #72237078, #72237090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 365773);