Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#385 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2666 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)(5,6,10,14,11,15,9,23)(7,12,19,27,17,26,18,33)(8,13,22,31,21,29,24,30)(16,25,20,35,32,36,34,28);(5,7)(6,12)(8,20,21,34)(9,18)(10,19)(11,17)(13,28,29,35)(14,27)(15,26)(16,22,32,24)(23,33)(25,30,36,31);(2,4)(5,8)(6,13)(7,16)(9,24)(10,22)(11,21)(12,25)(14,31)(15,29)(17,32)(18,34)(19,20)(23,30)(26,36)(27,35)(28,33);(7,17)(12,26)(16,32)(18,19)(20,34)(25,36)(27,33)(28,35)Polycyclic presentation:
2^9|a^2=h,b^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=di,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237178, #72237179, #72237188, … |
| maximal normal subgroup of | #72237178, #72237179, #72237188, … |
| maximal quotient of | #72239073, #72257466, #72257584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264942);