Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6367198 |
| 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 | |
1598 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)(3,12)(4,9)(5,10)(6,13)(7,11)(8,15)(14,16)(17,18,21,25)(19,26,28,31)(20,24,22,29)(23,30,32,27);(1,3,7,14)(2,8,11,5)(4,15,6,10)(9,16,13,12)(17,18)(19,23)(20,29)(21,25)(22,24)(26,27)(28,32)(30,31);(1,4,7,6)(2,9,11,13)(3,10,14,15)(5,16,8,12)(17,19)(18,23)(20,27)(21,28)(22,30)(24,31)(25,32)(26,29);(1,5)(2,10)(3,13)(4,16)(6,12)(7,8)(9,14)(11,15)(17,20,21,22)(18,24,25,29)(19,27,28,30)(23,31,32,26);(2,11)(3,14)(9,13)(10,15)(18,25)(23,32)(24,29)(26,31)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72283267, #72306907, #72306918, … |
| maximal normal subgroup of | #72283267, #72306907, #72306918, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274535);