Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498812 |
| 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 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,12)(7,17)(8,20)(9,10)(13,29)(14,27)(15,21)(16,28)(18,22)(19,25)(23,24)(26,32)(30,35)(31,33);(1,2)(3,4)(5,6)(7,13)(8,21)(9,16)(10,15)(11,14)(12,20)(17,33)(18,35)(19,30)(22,29)(23,27)(24,28)(25,32)(26,36)(31,34);(2,4)(5,7)(6,13)(8,22)(9,25)(10,17)(11,19)(12,26)(14,30)(15,33)(16,32)(18,24)(20,36)(21,29)(23,34)(27,31)(28,35);(6,14)(12,27)(13,30)(15,16)(21,28)(26,31)(29,35)(32,33)Polycyclic presentation:
2^9|e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,c^b=cg,d^b=di,e^b=eh,f^b=fi,h^b=hi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250368, #72250378, #72250383, … |
| maximal normal subgroup of | #72250368, #72250378, #72250383, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406027);