Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#2184 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10,13,18)(11,16,14,22)(12,15,21,24)(17,19,23,20);(10,15)(11,14)(12,20)(16,17)(18,24)(19,21)(22,23);(2,4);(5,7,8,6)(9,11,13,14)(10,16,18,22)(12,19,21,20)(15,23,24,17)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,e^b=eh,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72225017, #72225051, #72225063, … |
| maximal normal subgroup of | #72225017, #72225051, #72225063, … |
| maximal quotient of | #72224984, #72224985, #72224989, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402884);