Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1328 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)(13,14,16,18)(15,17,19,20)(21,22)(23,27)(24,31)(25,34)(26,29)(28,32)(30,36)(33,35);(5,7)(6,10)(8,11)(9,12)(13,15)(14,17)(16,19)(18,20)(21,23,26,25)(22,27,29,34)(24,32,33,30)(28,35,36,31);(5,8,9,6)(7,11,12,10)(15,19)(17,20)(21,24,25,32,26,33,23,30)(22,28,27,31,29,36,34,35);(2,4)(7,12)(10,11)(24,33)(28,36)(30,32)(31,35)Polycyclic presentation:
2^9|a^2=egi,b^2=i,c^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341151);