Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #433518 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15483 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
781 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,7,12,6,11)(3,13,21,23,15,8,16,24)(4,17,20,26,19,27,18,9)(14,28,30,25,29,32,31,22);(1,3,5,21,7,15,6,16)(2,8,11,23,12,13,10,24)(4,14,18,31,19,29,20,30)(9,25,17,32,26,22,27,28);(1,4)(2,9)(3,14,15,29)(5,20)(6,18)(7,19)(8,22,13,25)(10,17)(11,27)(12,26)(16,31,21,30)(23,32,24,28);(2,10)(3,15)(5,6)(8,23)(9,17)(11,12)(13,24)(14,29)(18,20)(22,32)(25,28)(26,27)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340714);