Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6,8,7)(9,10,14,18)(11,15,21,24)(12,16,13,17)(19,23,20,22);(1,2)(3,4)(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,22,17,23);(1,3,4,2)(10,16)(11,19)(12,13)(17,18)(20,21)(22,23);(5,7,8,6)(9,12,14,13)(10,16,18,17)(11,20,21,19)(15,22,24,23)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=e,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,g^b=gi,d^c=di,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72257937, #72264524, #72264531, … |
| maximal normal subgroup of | #72257937, #72264524, #72264531, … |
| maximal quotient of | #72265468, #72308396 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266275);