Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196110 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1611 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)(3,7)(4,6)(5,8)(9,10)(11,14,18,22)(12,16)(13,15)(17,21,20,24)(19,23);(1,3,4,8)(2,5,6,7)(11,17)(12,19)(14,21)(16,23)(18,20)(22,24);(9,11)(10,14,15,22)(12,20)(13,18)(16,24,23,21)(17,19);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fh,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72310622, #72310627, #72310639, … |
| maximal normal subgroup of | #72310622, #72310627, #72310639, … |
| maximal quotient of | #72314137, #72314190, C2×#374194 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281390);