Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #121899 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1445 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
695 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7,6,3,5,8)(9,10)(11,14,18,22)(12,19)(13,16)(15,20)(17,21,23,24);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14,16,22)(12,17)(13,18)(15,21,19,24)(20,23);(2,7)(3,8)(4,5)(9,12)(10,15,16,19)(11,17)(13,20)(14,21,22,24)(18,23)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,d^2=hi,e^2=hi,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,e^b=ei,f^b=fhi,g^b=gi,d^c=di,e^c=ei,f^c=fhi,g^c=ghi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29101);