Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13432 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2044 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C2 | #11174 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
763 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,8,14,7,13,6,12)(3,15,10,21,18,20,26,4)(5,11,25,31,24,30,23,16)(9,27,28,17,29,32,19,22);(1,3)(2,9)(4,16)(5,22)(6,26)(7,18)(8,10)(11,20)(12,28)(13,29)(14,19)(15,30)(17,24)(21,31)(23,32)(25,27);(1,4,5,19,7,21,24,28)(2,10,11,27,13,26,30,32)(3,16,17,12,18,31,22,14)(6,20,25,9,8,15,23,29)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,e^c=eh,f^c=fi,g^c=gh,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^e=gi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72281262 |
| maximal normal subgroup of | #72281262 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55672);