Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #200412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1685 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7,5,8,3,6)(9,10)(11,14,18,22)(12,19,20,15)(13,16)(17,21)(23,24);(2,6,8,7)(9,11)(10,14,16,22)(12,17)(13,18)(15,21,19,24)(20,23);(9,12,13,20)(10,15,16,19)(11,17,18,23)(14,21,22,24);(10,16)(14,22)(15,19)(21,24)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=i,e^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72260877 |
| maximal normal subgroup of | #72260877 |
| maximal quotient of | #72239058, #72261025 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107854);