Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1408 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)(3,4)(5,7)(6,8)(9,10,14,18,15,11,13,17)(12,19,20,24,21,16,22,23);(1,3,4,2,6,7,5,8)(9,11,14,17,15,10,13,18)(12,19,22,23,21,16,20,24);(1,4,6,5)(2,7,8,3)(9,12)(10,16)(11,19)(13,20)(14,22)(15,21)(17,23)(18,24);(1,4,6,5)(2,3,8,7)(9,13,15,14)(10,17,11,18)(12,20,21,22)(16,23,19,24)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=h,d^2=hi,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,f^b=fi,e^c=ei,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72281573, #72281602, #72281684 |
| maximal normal subgroup of | #72281573, #72281602, #72281684 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108747);