Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484095 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#509 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
473 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7,5,3,6,8)(10,14,17,22)(11,19)(12,20)(15,16,23,18)(21,24);(9,10,13,17)(11,15,21,23)(12,16,20,18)(14,19,22,24);(1,3,4,8,5,2,6,7)(9,11,20,24,13,21,12,19)(10,15,18,14,17,23,16,22);(2,3)(7,8)(9,12,13,20)(10,16,17,18)(11,19,21,24)(14,15,22,23)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dhi,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72216702, #72216759, #72216897, … |
| maximal normal subgroup of | #72216702, #72216759, #72216897, … |
| maximal quotient of | #72285862, #72285928, #72286138, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391306);