Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454375 |
| 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)(6,9)(10,12)(13,14,16,17)(15,18,19,20)(21,22,28,34)(23,31,24,36)(25,29,35,30)(26,33,27,32);(5,6,8,9)(7,10,11,12)(21,23,27,35,28,24,26,25)(22,29,33,36,34,30,32,31);(5,7)(6,10)(8,11)(9,12)(14,17)(18,20)(21,24,28,23)(22,30,34,29)(25,27,35,26)(31,33,36,32);(2,4)(7,11)(10,12)(13,15,16,19)(14,18,17,20)(21,25,26,24,28,35,27,23)(22,31,32,30,34,36,33,29)Polycyclic presentation:
2^9|a^2=fg,b^2=ei,c^2=i,d^2=f,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^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 369779);