Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19621 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#390 |
| prime radical | #60700 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7,5,8,3,6)(11,17)(12,19)(14,20)(16,22)(18,23)(21,24);(2,6,8,7)(9,10,13,15)(11,14,18,21)(12,16,19,22)(17,20,23,24);(9,11,12,17,13,18,19,23)(10,14,16,20,15,21,22,24);(10,15)(11,18)(16,22)(17,23)Polycyclic presentation:
2^9|a^2=g,b^2=egi,c^2=f,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72285558, #72322064, #72322139 |
| maximal normal subgroup of | #72285558, #72322064, #72322139 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 113620);