Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3334 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(6,8)(9,10,12,16,13,18,23,17)(11,19,21,15,22,14,20,24);(1,2)(3,6)(4,5)(7,8);(1,3,4,7)(2,6,5,8)(10,14)(11,20)(12,23)(15,16)(17,24)(18,19)(21,22);(2,5)(6,8)Polycyclic presentation:
2^9|a^2=g,c^2=e,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=de,f^c=fhi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72342431, #72342438, #72342442, … |
| maximal normal subgroup of | #72342431, #72342438, #72342442, … |
| maximal quotient of | #72343615, #72343661 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404751);