Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128061 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8101 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
1375 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)(3,13,16,26)(4,11,6,9)(5,19,21,10)(7,12)(8,15)(14,28)(17,32)(18,25)(20,27)(22,29,23,30)(24,31);(1,3,6,17,7,16,4,14)(2,8,11,23,12,24,9,22)(5,15,20,30,21,31,18,29)(10,26,27,32,19,13,25,28);(1,4,7,6)(2,9,12,11)(3,14,16,17)(5,20,21,18)(8,22,24,23)(10,27,19,25)(13,28,26,32)(15,30,31,29)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,f^2=i,b^a=bd,c^a=cg,d^a=dg,f^a=fh,g^a=gi,h^a=hi,d^b=dh,e^b=eg,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 | #72234494, #72234495, #72234497, … |
| maximal normal subgroup of | #72234494, #72234495, #72234497, … |
| maximal quotient of | #72234500, #72235162, #72235257, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35274);