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 | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1106 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)(7,10)(8,9)(11,12)(13,14)(15,18)(16,20)(17,19)(21,22)(23,27)(24,28)(25,29)(26,31)(30,34)(32,36)(33,35);(6,9)(10,12)(13,15,17,16)(14,18,19,20)(21,23,30,33,24,31,25,32)(22,26,34,36,28,27,29,35);(5,7)(6,10)(8,11)(9,12)(14,19)(18,20)(21,23,24,31)(22,27,28,26)(25,33,30,32)(29,35,34,36);(2,4)(7,11)(10,12)(13,16,17,15)(14,20,19,18)(23,31)(26,27)(32,33)(35,36)Polycyclic presentation:
2^9|b^2=fgi,c^2=i,d^2=f,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, 373162);