Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498821 |
| 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 | |
1538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,13)(8,14)(9,22)(10,27)(11,16)(12,18)(15,25)(17,32)(19,33)(20,34)(21,35)(23,29)(24,30)(26,28)(31,36);(5,7)(6,12)(8,17)(9,23)(10,20)(11,19)(13,22)(14,29)(15,34)(16,30)(18,25)(21,24)(26,36)(27,31)(28,33)(32,35);(2,4)(5,7)(6,13)(8,20)(9,24)(10,17)(11,19)(12,28)(14,34)(15,31)(16,33)(18,26)(21,23)(22,30)(25,36)(27,32)(29,35);(5,8,11,10)(6,14,16,27)(7,17,19,20)(9,21,26,25)(12,29,30,31)(13,32,33,34)(15,22,35,28)(18,23,24,36)Polycyclic presentation:
2^9|d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,g^a=gi,h^a=hi,c^b=cg,e^b=eh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231495, #72231524, #72231549, … |
| maximal normal subgroup of | #72231495, #72231524, #72231549, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406019);