Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,13,15)(11,16,12,14)(17,18,21,24)(19,22,20,23);(2,4)(5,6)(7,8)(9,11,13,12)(10,14,15,16)(18,22)(19,20)(23,24);(2,4)(6,8)(18,22)(19,20)(23,24);(9,12,13,11)(10,14,15,16)(18,23)(19,20)(22,24)Polycyclic presentation:
2^9|a^2=ef,b^2=efi,d^2=efi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72264680, #72264702, #72264747, … |
| maximal normal subgroup of | #72264680, #72264702, #72264747, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308327);