Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417442 |
| 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 | |
1318 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)(5,6,8,10)(7,9,11,12)(21,22,27,29,26,23,25,30)(24,31,34,36,33,28,32,35);(5,7,8,11)(6,9,10,12)(13,14)(15,18)(16,17)(19,20)(21,23,26,22)(24,28,33,31)(25,30,27,29)(32,35,34,36);(6,10)(9,12)(14,17)(18,20)(21,24,26,33)(22,28,23,31)(25,32,27,34)(29,35,30,36);(2,4)(13,15,16,19)(14,18,17,20)(22,29)(23,30)(25,27)(28,35)(31,36)(32,34)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=i,d^2=gi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356013);