Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476082 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1344 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)(5,6,9,11)(7,12,8,10)(13,14,16,18)(15,17,19,20)(21,22)(23,26)(24,27)(25,28)(29,36)(30,34)(31,33)(32,35);(5,7,9,8)(6,10,11,12)(13,15)(14,17)(16,19)(18,20)(21,23)(22,26)(24,30)(25,29)(27,34)(28,36)(31,32)(33,35);(5,8,9,7)(6,10,11,12)(14,18)(17,20)(22,27)(23,29)(25,32)(26,33)(30,31)(34,36);(2,4)(7,8)(10,12)(15,19)(17,20)(23,30)(26,34)(29,31)(33,36)Polycyclic presentation:
2^9|a^2=efi,b^2=e,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,g^b=gi,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, 383789);