Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394040 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1667 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)(11,17)(12,23)(13,20)(14,19)(15,18)(16,21)(22,24);(1,3,4,7)(2,5,6,8)(9,11,15,22)(10,16,18,23)(12,14,21,13)(17,19,24,20);(9,12,13,11,15,21,14,22)(10,17,20,16,18,24,19,23);(2,6)(5,8)(10,18)(16,23)(17,24)(19,20)Polycyclic presentation:
2^9|b^2=h,c^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72310607, #72310615, #72310657, … |
| maximal normal subgroup of | #72310607, #72310615, #72310657, … |
| maximal quotient of | C2×#409981 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 317177);