Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #180873 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#97 |
| prime radical | C22×D4 | M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,19,14,16,22,23,15,11,17,20,12,18,21,24);(1,3)(2,5)(4,8)(6,7)(10,16,11,18)(13,17)(19,24,20,23)(21,22);(3,8)(5,7)(9,11,13,20,14,18,22,24,15,10,17,19,12,16,21,23);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=i,f^2=h,h^2=i,b^a=be,d^a=deh,g^a=gi,c^b=cei,f^b=fi,g^c=gi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72239325, #72239465, #72239467, … |
| maximal normal subgroup of | #72239325, #72239465, #72239467, … |
| maximal quotient of | D4×#1375, #72309746, #72310063, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400514);