Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232215 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3053 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,18)(17,21)(20,24)(22,26)(23,25)(27,28);(9,12)(14,18)(16,24)(19,25)(20,21)(23,28)(26,27);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15)(14,19)(16,23)(17,22)(18,25)(20,27)(21,26)(24,28);(15,22)(19,26)(23,28)(25,27)Polycyclic presentation:
2^9|e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gi,c^b=cg,e^b=eh,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72235217, #72260828, #72261773, … |
| maximal normal subgroup of | #72235217, #72260828, #72261773, … |
| maximal quotient of | #72260952 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 141673);