Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,10)(11,12)(13,14,16,18)(15,19,20,17)(21,22)(23,26)(24,32)(25,29)(27,30)(28,34)(31,36)(33,35);(5,6,9,10)(7,11,8,12)(13,15,16,20)(14,17,18,19)(21,23,26,22,27,29,25,30)(24,33,35,32,36,28,34,31);(5,7,9,8)(6,11,10,12)(14,18)(17,19)(21,24,27,36)(22,28,30,33)(23,31,29,32)(25,35,26,34);(2,4)(5,8,9,7)(6,11,10,12)(15,20)(17,19)(22,29)(23,30)(25,26)(28,32)(31,33)(34,35)Polycyclic presentation:
2^9|a^2=fg,b^2=ei,c^2=ef,d^2=efi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 344547);