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 | |
4834 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)(9,10,13,18,14,16,12,17)(11,19,20,23,21,15,22,24);(2,4)(5,6)(7,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,19)(17,24)(18,23);(6,8)(10,16)(11,20)(12,13)(15,23)(19,24)(21,22);(11,21)(15,19)(20,22)(23,24)Polycyclic presentation:
2^9|a^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^b=di,g^b=gi,h^b=hi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72236779, #72237513, #72237517, … |
| maximal normal subgroup of | #72236779, #72237513, #72237517, … |
| maximal quotient of | #72236791 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264737);