Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1332 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)(5,6,8,10)(7,11,12,9)(13,14,16,17)(15,19,20,18)(21,22,26,30)(23,27,33,35)(24,29,25,28)(31,34,32,36);(5,7,8,12)(6,9,10,11)(14,17)(18,19)(21,23,26,33)(22,27,30,35)(24,32,25,31)(28,34,29,36);(2,4)(22,28)(23,31)(24,25)(29,30)(32,33)(34,36);(13,15,16,20)(14,18,17,19)(21,24,26,25)(22,28,30,29)(23,32,33,31)(27,34,35,36)Polycyclic presentation:
2^9|a^2=ei,b^2=eh,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323202);