Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #482357 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,8,5,7,3,6)(9,10)(11,17,19,24)(12,16)(13,15)(14,18,22,20)(21,23);(1,3,5,4)(2,6)(7,8)(9,11)(10,14)(12,20)(13,19)(15,22)(16,24)(17,23)(18,21);(2,7)(6,8)(10,15)(14,22)(16,23)(17,24);(2,7)(3,4)Polycyclic presentation:
2^9|a^2=f,b^2=gi,e^2=hi,f^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fghi,f^d=fi,f^e=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72240207, #72240210, #72240212 |
| maximal normal subgroup of | #72240207, #72240210, #72240212 |
| maximal quotient of | #72240579 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389640);