Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #482367 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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 | #34876 | C2 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
960 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(9,10,13,17,14,16,11,19)(12,15,20,24,21,18,22,23);(1,2)(3,6)(4,8)(5,7)(10,15)(11,13)(12,21)(16,18)(17,24)(19,23);(1,3,5,4)(2,6,7,8)(10,16)(15,18)(17,19)(23,24);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,20)(13,14)(15,23)(16,19)(18,24)(21,22)Polycyclic presentation:
2^9|a^2=f,c^2=hi,d^2=hi,e^2=h,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=fgh,f^d=fi,f^e=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72282313, #72282385, #72282486, … |
| maximal normal subgroup of | #72282313, #72282385, #72282486, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389866);