Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201928 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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×#3003 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(10,14)(11,18)(12,15)(17,22)(19,20)(21,24);(1,2)(3,5)(4,6)(7,8)(9,10,15,22,13,17,12,14)(11,16,18,24,19,23,20,21);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,20)(13,19)(14,21)(15,18)(17,23)(22,24);(11,19)(16,23)(18,20)(21,24)Polycyclic presentation:
2^9|a^2=g,b^2=egh,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,f^b=fi,d^c=dh,e^c=ei,f^c=fh,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72308356, C2×#201929 |
| maximal normal subgroup of | #72308356, C2×#201929 |
| maximal quotient of | #72308391, #72308394, C2×#201929, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109125);