Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128065 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8101 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
1415 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10)(3,11,13,28)(4,15,18,27)(6,8,19,24)(7,21,23,16)(9,20,25,17)(12,22,30,26)(14,29,31,32);(1,3)(2,7,8,22)(4,16)(5,13,19,31)(6,14)(9,11,15,29)(10,26)(12,17)(18,21,20,30)(23,24)(25,32)(27,28);(2,8)(4,17)(5,19)(7,22)(12,16)(13,31)(25,27)(28,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=i,b^a=bd,c^a=cg,d^a=dg,f^a=fhi,g^a=gi,h^a=hi,d^b=dh,e^b=egi,g^b=gi,d^c=di,e^c=ei,e^d=ei,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72235132, #72235133, #72235134, … |
| maximal normal subgroup of | #72235132, #72235133, #72235134, … |
| maximal quotient of | #72235162, #72235167, #72235252, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35275);