Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#390 |
| prime radical | C2×#3050 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1316 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)(3,5,7,8)(4,6)(9,10,13,17,14,18,12,15)(11,19,22,24,20,23,21,16);(1,3)(2,5,6,8)(4,7)(10,15,18,17)(11,20)(16,19,24,23)(21,22);(3,7)(5,8)(9,11,12,21,14,20,13,22)(10,16,17,19,18,24,15,23);(9,12,14,13)(10,15,18,17)(11,21,20,22)(16,23,24,19)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=f,d^2=hi,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,c^b=ch,e^b=ei,g^b=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72279957, #72279972, #72280040, … |
| maximal normal subgroup of | #72279957, #72279972, #72280040, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 113224);