Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×#120 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,7,8)(9,10,15,19)(11,20,22,24)(12,21,16,17)(13,18,14,23);(9,11,14,21,15,22,13,17)(10,16,18,20,19,12,23,24);(1,2,3,4)(9,12,15,16)(10,17,19,21)(11,18,22,23)(13,24,14,20);(9,13,15,14)(10,18,19,23)(11,17,22,21)(12,24,16,20)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,g^b=gi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72240012 |
| maximal normal subgroup of | #72240012 |
| maximal quotient of | #72240021 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265643);