Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3543 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5,6,7)(9,10,13,16,14,17,12,18)(11,19,20,23,21,15,22,24);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24);(3,8)(5,7)(10,16)(12,13)(15,23)(17,18)(19,24)(20,22);(10,17)(15,19)(16,18)(23,24)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,e^c=ei,f^c=fh,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72226694, #72258074, #72284648, … |
| maximal normal subgroup of | #72226694, #72258074, #72284648, … |
| maximal quotient of | #72224609 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94659);