Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6368846 |
| 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 | |
1576 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)(3,13,14,9)(4,10,15,16)(5,11,6,8)(17,18,20,25)(19,24,27,22)(21,23,29,26)(28,31,32,30);(2,8)(4,9)(5,6)(10,16)(11,12)(13,15)(17,19,20,27)(18,22,25,24)(21,28,29,32)(23,30,26,31);(1,3,7,14)(2,9,12,13)(4,11,15,8)(5,16,6,10)(18,23)(21,29)(22,30)(24,31)(25,26)(28,32);(1,4)(2,10)(3,11)(5,9)(6,13)(7,15)(8,14)(12,16)(17,19)(18,24)(20,27)(21,28)(22,25)(23,31)(26,30)(29,32);(1,5,7,6)(2,8,12,11)(3,10,14,16)(4,9,15,13)(19,27)(22,24)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^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,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72306584, #72306650, #72306653, … |
| maximal normal subgroup of | #72306584, #72306650, #72306653, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6276114);