Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2358 |
| prime radical | C2×H(5,𝔽2) | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,3,4,6,8,7)(9,10,14,17,15,19,13,18)(11,12,20,16,21,23,24,22);(1,3,5,6,4,7,8,2)(9,11,14,20,15,21,13,24)(10,16,17,23,19,22,18,12);(2,6)(3,7)(9,12,15,23)(10,11,19,21)(13,22,14,16)(17,20,18,24);(9,11)(10,12)(13,24)(14,20)(15,21)(16,17)(18,22)(19,23)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ci,d^b=dei,e^b=eh,d^c=dh,f^c=fh,e^d=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 134384);