Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393940 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
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 | |
1054 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)(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,18)(17,19)(21,22)(23,29)(24,27)(25,28)(26,31)(30,34)(32,36)(33,35);(6,9)(10,12)(13,15,17,16)(14,18,19,20)(21,23,24,31)(22,26,27,29)(25,30,33,32)(28,36,35,34);(1,2,4,3)(5,7,8,11)(6,10,9,12)(14,19)(18,20)(22,27)(23,30)(25,33)(26,34)(29,36)(31,32);(1,3,4,2)(13,16,17,15)(14,18,19,20)Polycyclic presentation:
2^9|b^2=fh,c^2=ef,d^2=fi,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 316182);