Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417442 |
| 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 | |
1250 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)(15,19,20,17)(21,22)(23,29)(24,27)(25,28)(26,31)(30,34)(32,36)(33,35);(5,7)(6,10)(8,11)(9,12)(13,15,16,20)(14,17,18,19)(21,23)(22,26)(24,31)(25,30)(27,29)(28,36)(32,33)(34,35);(5,8,9,6)(7,11,12,10)(14,18)(17,19)(22,27)(23,30)(25,33)(26,34)(29,36)(31,32);(2,4)(7,12)(10,11)(15,20)(17,19)(23,31)(26,29)(30,32)(34,36)Polycyclic presentation:
2^9|a^2=ei,b^2=fi,c^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 333119);