Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129405 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10)(11,19)(12,16)(13,22,23,17)(14,18)(15,20,24,21);(5,6)(7,8)(9,11,14,12)(10,15,18,24)(13,20,23,21)(16,17,19,22);(2,4)(6,8)(9,12,14,11)(10,16,18,19)(13,21,23,20)(15,17,24,22)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,g^b=gh,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72236757, #72240229, #72256600 |
| maximal normal subgroup of | #72236757, #72240229, #72256600 |
| maximal quotient of | #72236760 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54754);