Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #197339 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#2619 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1621 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7,6,3,4,8)(9,10)(11,17)(12,20,21,15)(13,19)(14,16)(18,22,24,23);(1,3,6,2)(4,8,5,7)(9,11,17,10,14,19,13,16)(12,18,22,15,21,24,23,20);(9,12)(10,15)(11,18)(13,23)(14,21)(16,20)(17,22)(19,24);(1,4,6,5)(2,7,3,8)(12,21)(15,20)(18,24)(22,23)Polycyclic presentation:
2^9|a^2=g,b^2=eghi,d^2=i,e^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214356, #72214667, #72214750 |
| maximal normal subgroup of | #72214356, #72214667, #72214750 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 104668);