Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514160 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(5,6)(7,16)(8,20)(9,14)(10,13)(11,18)(12,23)(15,24)(17,30)(19,29)(21,27)(22,32)(25,28)(26,31)(33,35)(34,36);(1,2,4,3)(5,7,9,18)(6,11,14,16)(8,21,22,35)(10,25,24,19)(12,30,31,36)(13,29,15,28)(17,23,34,26)(20,33,32,27);(5,8)(6,12)(7,17)(9,22)(10,26)(11,27)(13,20)(14,31)(15,32)(16,33)(18,34)(19,21)(23,24)(25,35)(28,36)(29,30);(6,13)(10,24)(11,28)(12,20)(14,15)(16,29)(19,25)(21,35)(23,26)(27,36)(30,33)(31,32);(1,3,4,2)(6,14)(11,16)(12,31)(13,15)(20,32)(27,33)(28,29)(30,36)Polycyclic presentation:
2^9|b^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7425992);