Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| 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 | |
1136 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)(15,17)(16,18)(19,20)(21,22,25,30,24,29,26,28)(23,31,32,36,33,27,34,35);(5,7)(6,10)(8,11)(9,12)(13,15,16,19)(14,17,18,20)(21,23,26,34,24,33,25,32)(22,27,28,36,29,31,30,35);(5,8,9,6)(7,11,12,10)(14,18)(17,20)(22,28)(23,32)(25,26)(29,30)(33,34)(35,36);(2,4)(7,12)(10,11)(22,29)(23,33)(28,30)(32,34)Polycyclic presentation:
2^9|a^2=eg,b^2=fi,c^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^c=fi,g^c=gi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340121);