Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(21,22,24,28)(23,26,30,33)(25,27,32,29)(31,34,36,35);(6,9)(10,12)(13,14,16,18)(15,19,20,17)(21,23,24,30)(22,26,28,33)(25,31,32,36)(27,34,29,35);(2,4)(13,15,16,20)(14,17,18,19)(23,30)(26,33)(31,36)(34,35);(5,7,8,11)(6,10,9,12)(22,27)(25,32)(26,34)(28,29)(31,36)(33,35)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gi,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 310611);