Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23882 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14171 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,11,5,19)(3,12,14,8,15,24,20,22)(4,16,17,9,18,26,21,25)(13,23,30,32,29,31,28,27);(1,3)(2,8)(4,13)(5,20)(6,14)(7,15)(9,23)(10,24)(11,22)(12,19)(16,27)(17,30)(18,29)(21,28)(25,31)(26,32);(8,22)(12,24)(23,31)(27,32);(1,4,6,17,7,18,5,21)(2,9,10,26,11,25,19,16)(3,13,14,30,15,29,20,28)(8,23,24,32,22,31,12,27)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215995, #72216047, #72216166, … |
| maximal normal subgroup of | #72215995, #72216047, #72216166, … |
| maximal quotient of | C2×#375830, #72252151, #72256352, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283026);