Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #298939 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C4×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #25721 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1111 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)(3,7)(4,6)(5,8)(17,18,21,24,20,23,22,19);(1,3,4,8)(2,5,6,7)(9,10,13,14,12,16,11,15)(17,19,21,18,20,24,22,23);(10,14)(11,13)(15,16)(18,19)(21,22)(23,24);(2,6)(5,7)(18,23)(19,24)Polycyclic presentation:
2^9|a^2=f,b^2=eghi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72226827 |
| maximal normal subgroup of | #72226827 |
| maximal quotient of | #72226815 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 226857);