Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
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 | |
1246 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)(5,6,9,8)(7,11,12,10)(13,14)(15,19)(16,18)(17,20)(21,22,26,30)(23,32,34,28)(24,35,33,36)(25,29,27,31);(5,7,9,12)(6,10,8,11)(13,15,16,20)(14,17,18,19)(21,23,26,34)(22,28,30,32)(24,27,33,25)(29,36,31,35);(5,8,9,6)(7,11,12,10)(14,18)(17,19)(21,24,25,23,26,33,27,34)(22,28,31,36,30,32,29,35);(2,4)(7,12)(10,11)(15,20)(17,19)(21,25,26,27)(22,29,30,31)(23,33,34,24)(28,35,32,36)Polycyclic presentation:
2^9|a^2=efg,b^2=e,c^2=ef,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342005);