Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439025 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
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 | |
4952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,12,16)(7,10,15,11)(8,17,22,27)(9,14,25,30)(13,21,28,20)(18,23,29,24)(19,31,34,35)(26,33,36,32);(1,2,3,4)(5,7,11,6,12,15,10,16)(8,13,20,17,22,28,21,27)(9,14,23,18,25,30,24,29)(19,31,33,26,34,35,32,36);(5,8,11,21,12,22,10,20)(6,13,15,27,16,28,7,17)(9,19,24,33,25,34,23,32)(14,26,29,35,30,36,18,31);(5,9)(6,14)(7,18)(8,19)(10,23)(11,24)(12,25)(13,26)(15,29)(16,30)(17,31)(20,32)(21,33)(22,34)(27,35)(28,36);(5,10,12,11)(6,15,16,7)(8,20,22,21)(9,23,25,24)(13,27,28,17)(14,29,30,18)(19,32,34,33)(26,35,36,31)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350487);