Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #334695 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #25739 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
804 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,6,7,5,3,4,8)(9,10)(11,19)(12,15)(13,17)(14,18)(16,20)(21,23)(22,24);(1,3,5,2)(4,8,6,7)(9,10)(11,16)(12,17)(13,15)(14,18)(19,20)(21,24)(22,23);(2,3)(7,8)(10,15)(12,13)(16,19)(17,18)(20,22)(23,24);(2,7)(3,8)(4,6)(9,11,14,21)(10,16,18,24)(12,22,13,20)(15,19,17,23)Polycyclic presentation:
2^9|a^2=eh,b^2=h,d^2=i,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=gi,f^c=fi,g^c=gi,e^d=eh,f^d=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 241973);