Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8101 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#95 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,19,21,15)(12,17)(13,18)(14,16)(20,24,22,23);(1,3)(2,5,6,7)(4,8)(9,11,13,22,14,21,12,20)(10,15,18,24,16,19,17,23);(10,16)(15,19)(17,18)(23,24);(9,12,14,13)(10,17,16,18)(11,20,21,22)(15,23,19,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,g^2=h,b^a=be,c^a=ch,e^a=eh,g^a=gi,e^b=ei,f^b=fh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72235080, #72235086, #72235091, … |
| maximal normal subgroup of | #72235080, #72235086, #72235091, … |
| maximal quotient of | #72284838, C2×#181378, #72309128, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88574);