Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393940 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1496 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)(15,17)(16,18)(19,20)(23,31)(24,25)(27,35)(28,29)(32,36)(33,34);(5,7,9,8)(6,10,11,12)(13,15,16,19)(14,17,18,20)(21,22,26,30)(23,32,34,35)(24,28,25,29)(27,31,36,33);(2,4)(5,8,9,7)(6,10,11,12)(21,23)(22,27)(24,33)(25,31)(26,34)(28,35)(29,32)(30,36);(14,18)(17,20)(21,24,26,25)(22,28,30,29)(23,33,34,31)(27,35,36,32)Polycyclic presentation:
2^9|a^2=e,b^2=eh,c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312720);