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)(7,16)(8,19)(9,10)(13,27)(14,31)(15,26)(17,23)(18,33)(20,24)(21,22)(25,29)(28,34)(30,36);(1,2)(3,4)(5,6)(7,13)(8,20)(9,14)(10,24)(11,15)(12,19)(16,29)(17,28)(18,34)(21,26)(22,31)(23,30)(25,32)(27,33)(35,36);(2,4)(5,7)(6,13)(8,18)(9,23)(10,16)(11,17)(12,25)(14,30)(15,28)(19,32)(20,34)(21,35)(22,33)(24,29)(26,36)(27,31);(7,17)(13,28)(16,23)(18,33)(25,36)(27,34)(29,30)(32,35)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,e^b=eh,h^b=hi,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250059, #72250071, #72250075, … |
| maximal normal subgroup of | #72250059, #72250071, #72250075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406008);