Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1404 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)(3,4)(5,6,8,7)(9,10)(11,13)(12,15)(14,19)(16,20)(17,21)(18,23)(22,24);(9,11,14,21,16,22,15,18)(10,17,12,13,20,23,19,24);(2,4)(9,12,14,20,16,19,15,10)(11,17,21,13,22,23,18,24);(5,7,8,6)(9,13,14,23,16,24,15,17)(10,18,12,11,20,21,19,22)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,d^2=eh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=di,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216143 |
| maximal normal subgroup of | #72216143 |
| maximal quotient of | #72216072, #72216075, #72216101 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283493);