Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(6,9)(10,12)(13,14,16,17)(15,18,19,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,34)(32,36)(33,35);(1,2,4,3)(5,6,8,9)(7,10,11,12)(21,23,30,33,24,31,25,32)(22,26,34,36,27,29,28,35);(2,3)(7,11)(10,12)(14,17)(18,20)(21,23,24,31)(22,26,27,29)(25,33,30,32)(28,36,34,35);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,15,16,19)(14,18,17,20)(22,27)(23,31)(28,34)(32,33)Polycyclic presentation:
2^9|a^2=fi,b^2=eghi,c^2=i,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380325);