Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431785 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
7100 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)(3,4)(5,6)(7,17)(8,14)(9,15)(10,13)(11,16)(12,18)(19,31)(20,27)(21,26)(22,29)(23,30)(24,28)(25,32)(33,36)(34,35);(2,4)(5,7)(6,12)(8,19)(9,20)(10,18)(11,21)(13,26)(14,25)(15,17)(16,27)(22,33)(23,34)(24,32)(28,36)(29,35)(30,31);(6,13)(7,18)(9,10)(14,28)(15,16)(17,26)(19,32)(20,21)(23,24)(29,30)(31,36)(33,34);(5,8)(6,14)(7,19)(9,23)(10,24)(11,22)(12,25)(13,28)(15,30)(16,29)(17,31)(18,32)(20,34)(21,33)(26,36)(27,35);(8,22)(14,29)(19,33)(23,24)(25,35)(28,30)(31,36)(32,34)Polycyclic presentation:
2^9|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,f^b=fi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338981);