Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19279 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#390 |
| prime radical | C2×#3147 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1289 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7,6,3,4,8)(9,10,13,17,14,18,12,15)(11,19,22,24,20,23,21,16);(1,3,6,2)(4,8,5,7)(10,15,18,17)(11,20)(16,19,24,23)(21,22);(9,11,12,21,14,20,13,22)(10,16,17,19,18,24,15,23);(1,4,6,5)(2,7,3,8)Polycyclic presentation:
2^9|a^2=g,b^2=egi,c^2=f,d^2=i,e^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,c^b=ch,e^b=ei,g^b=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72310170, #72310174, #72310175, … |
| maximal normal subgroup of | #72310170, #72310174, #72310175, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112693);