Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×D42 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,23,21,20,19,24);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(17,19,21,22)(18,23,20,24);(10,14)(11,15)(18,20)(23,24);(3,8)(5,7)(10,14)(13,16)(17,20,22,24,21,18,19,23);(2,6)(5,7)(18,20)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cgi,d^b=df,d^c=di,e^d=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264731 |
| maximal normal subgroup of | #72264731 |
| maximal quotient of | #72264750 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6329286);