Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501172 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1057 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)(10,14)(15,20)(16,21)(22,24);(1,3,4,8)(2,5,6,7)(11,17)(12,19)(15,22)(20,24);(9,10,12,14)(11,15,18,20)(13,16,19,21)(17,22,23,24);(2,6)(5,7)(9,11,13,17)(10,15,16,22)(12,18,19,23)(14,20,21,24)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^c=gh,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72285788, #72285851, #72286061, … |
| maximal normal subgroup of | #72285788, #72285851, #72286061, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409906);