Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4838 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)(3,4)(5,6)(7,8)(9,10,13,17,14,15,12,20)(11,16,18,23,19,22,21,24);(2,4)(10,15)(11,18)(12,13)(16,23)(19,21)(22,24);(6,8)(9,11)(10,16)(12,21)(13,18)(14,19)(15,22)(17,23)(20,24);(11,19)(16,22)(18,21)(23,24)Polycyclic presentation:
2^9|a^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=di,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72236779, #72237178, #72237181, … |
| maximal normal subgroup of | #72236779, #72237178, #72237181, … |
| maximal quotient of | #72236786, #72239074, #72239075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264777);