Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460565 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16367 |
| 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 | |
969 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,9,20,26)(3,7,13,23,12,22,18,27)(4,14,16,28,17,8,19,25)(11,29,15,32,30,21,31,24);(1,3)(2,7,9,22)(4,15)(5,18)(6,12)(8,24,14,32)(10,27,26,23)(11,16)(13,20)(17,31)(19,30)(21,28,29,25);(1,4)(2,8)(3,11)(5,19)(6,17)(7,21)(9,14)(10,28)(12,30)(13,31)(15,18)(16,20)(22,29)(23,32)(24,27)(25,26);(2,9)(5,20)(7,22)(11,30)(13,18)(16,19)(21,29)(25,28)Polycyclic presentation:
2^9|a^2=g,b^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,g^c=gi,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72287267, #72287280, #72287354, … |
| maximal normal subgroup of | #72287267, #72287280, #72287354, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367761);