Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #61234 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#436 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1675 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,18,15,19,13,29)(11,20,23,31,24,36,22,16)(12,17,28,34,25,32,27,40)(21,35,38,33,26,39,37,30);(1,3)(2,5)(4,8)(6,7)(12,25)(17,32)(21,26)(27,28)(30,33)(34,40)(35,39)(37,38);(9,11,13,22,15,24,14,23)(10,16,18,20,19,31,29,36)(12,26,27,38,25,21,28,37)(17,33,34,39,32,30,40,35);(3,8)(5,7)(12,25)(17,32)(21,26)(27,28)(30,33)(34,40)(35,39)(37,38);(2,6)(5,7)(9,12,14,28,15,25,13,27)(10,17,18,34,19,32,29,40)(11,21,23,38,24,26,22,37)(16,30,20,35,31,33,36,39)Polycyclic presentation:
2^9|a^2=h,c^2=g,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,e^c=ei,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72313678, #72313701, #72313717, … |
| maximal normal subgroup of | #72313678, #72313701, #72313717, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6310525);