Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573635 |
| 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 | |
2554 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)(7,19)(8,15)(9,16)(10,17)(11,14)(12,18)(13,22)(20,29)(21,32)(23,31)(24,36)(25,33)(26,30)(27,34)(28,35);(1,2,3,4)(5,7)(6,13)(8,11)(9,24)(10,22)(12,20)(14,29)(15,18)(16,31)(17,19)(21,26)(23,25)(27,35)(28,30)(32,34)(33,36);(6,14)(7,20)(8,22)(11,12)(15,29)(16,30)(17,18)(21,24)(23,35)(26,27)(31,36)(33,34);(5,8)(6,15)(7,12)(9,21)(10,20)(11,22)(13,14)(16,32)(17,29)(18,19)(23,26)(24,27)(25,35)(28,33)(30,31)(34,36);(5,9)(6,16)(7,21)(8,23)(10,25)(11,26)(12,27)(13,28)(14,30)(15,31)(17,33)(18,34)(19,32)(20,24)(22,35)(29,36)Polycyclic presentation:
2^9|b^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481423);