Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499754 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#435 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17)(8,22)(9,24)(10,28)(11,16)(12,19)(13,23)(14,27)(15,18)(20,32)(21,36)(25,30)(26,33)(29,35)(31,34);(1,2)(3,4)(5,7,11,20)(6,12,16,29)(8,18,21,10)(9,25,26,23)(13,15,30,28)(14,22,31,36)(17,33,32,24)(19,34,35,27);(2,4)(9,26)(14,31)(17,32)(19,35)(22,36)(23,25)(24,33)(27,34);(5,8)(6,13)(7,18)(9,19)(10,20)(11,21)(12,15)(14,32)(16,30)(17,31)(22,24)(23,27)(25,34)(26,35)(28,29)(33,36)Polycyclic presentation:
2^9|b^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=eh,f^b=fi,h^b=hi,e^c=ei,e^d=eh,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406952);