Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421767 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15297 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,6,11,5,21)(3,12,15,8,13,25,16,23)(4,17,19,9,20,28,22,27)(14,31,32,29,30,24,18,26);(1,3,7,15,6,13,5,16)(2,8,10,25,11,23,21,12)(4,18,19,14,20,32,22,30)(9,26,28,31,27,29,17,24);(3,13)(5,7)(8,23)(10,21)(14,30)(17,28)(19,22)(24,31);(1,4)(2,9)(3,14)(5,22)(6,20)(7,19)(8,24)(10,28)(11,27)(12,29)(13,30)(15,32)(16,18)(17,21)(23,31)(25,26)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72287080, #72287261, #72287267, … |
| maximal normal subgroup of | #72287080, #72287261, #72287267, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328963);