Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #234851 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3006 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2312 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)(3,5)(4,6)(7,8)(10,15)(11,18)(12,13)(17,21)(20,22)(23,24);(5,8)(9,10)(11,19)(12,21)(13,15)(14,17)(16,20)(18,23)(22,24);(1,3)(2,5)(4,7)(6,8)(9,11,14,20)(10,16,17,19)(12,18,13,22)(15,23,21,24);(3,7)(5,8)(11,20)(16,19)(18,22)(23,24)Polycyclic presentation:
2^9|c^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,d^c=dhi,e^c=ei,f^c=fh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72257736, #72257749, #72263963 |
| maximal normal subgroup of | #72257736, #72257749, #72263963 |
| maximal quotient of | #72264388, #72264414 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142049);