Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480482 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×D4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2168 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)(7,15,18,29)(8,12)(9,13,22,27)(10,14)(11,17)(16,28,30,36)(19,25)(20,26,32,35)(21,24)(23,31)(33,34);(5,7,9,21)(6,11,13,15)(8,16,20,33)(10,18,22,17)(12,23,26,28)(14,24,27,29)(19,30,32,31)(25,34,35,36);(2,4)(5,8)(6,12)(7,16)(9,20)(10,19)(11,23)(13,26)(14,25)(15,28)(17,31)(18,30)(21,33)(22,32)(24,34)(27,35)(29,36);(8,19)(12,25)(16,30)(20,32)(23,34)(26,35)(28,36)(31,33)Polycyclic presentation:
2^9|a^2=g,b^2=egi,e^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,d^c=di,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72317600, #72317603, #72317607, … |
| maximal normal subgroup of | #72317600, #72317603, #72317607, … |
| maximal quotient of | #72317653, #72317692 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401914);