Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6365322 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1689 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,12)(3,4,10,9)(5,11,6,8)(13,16,15,14)(17,18,21,25)(19,22,26,30)(20,23,28,24)(27,29,32,31);(2,8)(3,10)(4,14)(5,6)(9,16)(11,12)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(1,3,7,10)(2,9,12,4)(5,15,6,13)(8,14,11,16)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(1,4,7,9)(2,10,12,3)(5,14,6,16)(8,15,11,13);(1,5,7,6)(2,8,12,11)(3,13,10,15)(4,14,9,16)(19,26)(22,30)(27,32)(29,31)Polycyclic presentation:
2^9|a^2=hi,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh71 conjugacy classes — view the full character table.
| maximal subgroup of | #72338427, #72338434, #72338440 |
| maximal normal subgroup of | #72338427, #72338434, #72338440 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6272557);