Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393899 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#392 |
| prime radical | #34737 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1889 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)(9,10)(11,18)(12,23)(13,16)(14,17)(15,20)(19,21)(22,24);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,22)(13,20)(14,21)(16,19)(17,24)(18,23);(9,10)(11,19,21,18)(12,23)(13,16)(14,17)(15,20,24,22);(2,6)(5,8)Polycyclic presentation:
2^9|b^2=h,c^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72237198, #72249560, #72252472, … |
| maximal normal subgroup of | #72237198, #72249560, #72252472, … |
| maximal quotient of | #72237212, #72237253, #72257564, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 313441);