Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439035 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2679 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6)(7,10)(8,18)(9,13)(11,15)(12,23)(14,20)(16,24)(17,26)(19,22)(21,32)(25,27)(28,30)(29,35)(31,36)(33,34);(2,3)(5,7,10,6)(8,19,22,18)(9,25,26,30)(11,15,20,14)(12,24,16,23)(13,28,17,27)(21,31,34,29)(32,35,33,36);(1,2,4,3)(5,8,20,24,10,22,11,23)(6,12,15,19,7,16,14,18)(9,21,28,36,26,34,27,35)(13,29,25,33,17,31,30,32);(1,3,4,2)(5,9,10,26)(6,13,7,17)(8,21,22,34)(11,27,20,28)(12,29,16,31)(14,30,15,25)(18,32,19,33)(23,35,24,36);(8,22)(12,16)(18,19)(21,34)(23,24)(29,31)(32,33)(35,36)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fg,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351504);