Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7943 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#95 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C26 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2878 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)(3,7,8,5)(4,6)(9,10,12,16,14,18,13,17)(11,19,21,15,20,24,22,23);(1,3)(2,5)(4,8)(6,7)(9,11,13,22,14,20,12,21)(10,15,16,24,18,23,17,19);(3,8)(5,7)(11,20)(15,23)(19,24)(21,22);(9,12,14,13)(10,16,18,17)(11,21,20,22)(15,24,23,19)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,e^2=hi,f^2=hi,g^2=hi,b^a=be,e^a=eh,g^a=ghi,c^b=ch,f^b=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72235627, #72236746, #72236791, … |
| maximal normal subgroup of | #72235627, #72236746, #72236791, … |
| maximal quotient of | #72239123, #72308433, C2×#181326, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88522);