Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23900 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#379 |
| prime radical | #62773 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2673 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(1,2)(3,5)(4,6)(7,8);(11,17)(14,20)(18,23)(21,24)Polycyclic presentation:
2^9|a^2=g,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=di,f^b=fh,g^b=gh,e^c=eh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72238180, #72238182, #72238184, … |
| maximal normal subgroup of | #72238180, #72238182, #72238184, … |
| maximal quotient of | C2×#184919, #72338847 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92115);