Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #267371 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #2408 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1179 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)(11,17)(14,23)(18,24)(19,22);(1,3)(2,5)(4,8)(6,7)(9,10,12,15)(11,18,17,24)(13,21,20,16)(14,22,23,19);(1,2)(3,5)(4,6)(7,8)(9,11,12,17)(10,14,15,23)(13,22,20,19)(16,24,21,18);(1,3)(2,5)(4,8)(6,7)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gi,e^c=eh,g^c=gi,f^d=fh59 conjugacy classes — view the full character table.
| maximal subgroup of | #72339479, #72339480, #72339484, … |
| maximal normal subgroup of | #72339479, #72339480, #72339484, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 174595);