Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #267419 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #61707 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1029 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(9,10)(11,19,22,15)(12,21)(13,20,24,17)(14,18)(16,23);(1,2)(3,7)(4,6)(5,8)(9,11,14,22)(10,15,18,19)(12,20,23,17)(13,21,24,16);(2,6)(3,4)(7,8)(9,12)(10,16)(11,20)(13,15)(14,23)(17,22)(18,21)(19,24);(1,3,5,4)(2,7,8,6)(9,13)(10,17)(11,21)(12,15)(14,24)(16,22)(18,20)(19,23)Polycyclic presentation:
2^9|a^2=fh,b^2=h,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gi,d^c=di,e^c=eh,g^c=gi,f^d=fh53 conjugacy classes — view the full character table.
| maximal subgroup of | #72268291, #72268293, #72268318, … |
| maximal normal subgroup of | #72268291, #72268293, #72268318, … |
| maximal quotient of | #72313406 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 174634);