Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6365522 |
| 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 | |
1573 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,9)(3,13,14,15)(4,10,8,16)(5,12,6,11)(17,18)(19,22)(20,28)(21,25)(23,27)(24,29)(26,31)(30,32);(1,3)(2,8)(4,9)(5,10)(6,16)(7,14)(11,13)(12,15)(17,19,22,18)(20,26,30,24)(21,25,23,27)(28,29,32,31);(2,9)(3,14)(10,16)(11,12)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28);(1,4)(2,10)(3,12)(5,15)(6,13)(7,8)(9,16)(11,14)(18,19)(24,26)(25,27)(28,32);(1,5,7,6)(2,11,9,12)(3,10,14,16)(4,15,8,13)(17,21,22,23)(18,25,19,27)(20,29,30,31)(24,32,26,28)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ei,f^b=fh,g^c=gi,f^d=fh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72283125, #72306896, #72306900, … |
| maximal normal subgroup of | #72283125, #72306896, #72306900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6272880);