Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(6,7)(9,10,13,19,15,20,14,18)(11,16,22,21,23,12,17,24);(5,6,8,7)(9,11)(10,16)(12,20)(13,22)(14,17)(15,23)(18,24)(19,21);(2,4)(5,6,8,7)(9,10)(11,21)(12,22)(13,19)(14,18)(15,20)(16,17)(23,24);(5,7,8,6)(9,12,13,24,15,16,14,21)(10,17,19,11,20,22,18,23)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,d^2=eh,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216201 |
| maximal normal subgroup of | #72216201 |
| maximal quotient of | #72215970, #72216074, #72216075 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283487);