Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| 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 | |
5006 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)(11,15)(12,14)(13,17)(16,21)(18,24)(19,23)(20,22);(10,14)(11,18)(12,21)(16,17)(19,20)(22,24);(2,4)(9,11)(10,15)(12,20)(13,19)(14,22)(16,24)(17,23)(18,21);(11,19)(15,23)(18,20)(22,24)Polycyclic presentation:
2^9|a^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,e^b=ei,g^b=gi,d^c=di,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72235935, #72236700, #72236744, … |
| maximal normal subgroup of | #72235935, #72236700, #72236744, … |
| maximal quotient of | #72235941, #72239034, #72281497, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264767);