Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #244927 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1052 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)(5,6,11,14)(7,15,9,18)(8,13,10,19)(12,17,20,16)(23,31)(24,25)(27,35)(29,30)(32,36)(33,34);(5,7,11,9)(6,12,14,20)(8,16,10,17)(13,18,19,15)(21,22)(23,32)(24,30)(25,29)(26,28)(27,33)(31,36)(34,35);(2,4)(5,8,11,10)(6,13,14,19)(7,16,9,17)(12,18,20,15)(21,23,26,34)(22,27,28,36)(24,33,25,31)(29,32,30,35);(5,9,11,7)(6,12,14,20)(8,17,10,16)(13,18,19,15)(22,28)(27,36)(29,30)(32,35)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=hi,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=deh,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^b=eh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152192);