Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9307 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #60561 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6,5,3,8,7)(9,10,13,17,14,18,12,16)(11,19,22,23,20,24,21,15);(1,3)(2,5)(4,6)(7,8)(9,11)(10,15,18,23)(12,21)(13,22)(14,20)(16,24,17,19);(1,2)(3,5)(4,6)(7,8)(11,20)(15,23)(19,24)(21,22);(9,12,14,13)(10,16,18,17)(11,21,20,22)(15,24,23,19)Polycyclic presentation:
2^9|a^2=g,b^2=egi,d^2=h,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,e^b=ei,g^b=gi,f^c=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106630);