Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6348967 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2967 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)(3,11)(4,8)(5,9)(6,10)(7,12)(13,15)(14,16)(17,18)(19,28)(20,27)(21,25)(22,26)(23,30)(24,29)(31,32);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19)(18,23)(20,29)(21,30)(22,31)(24,26)(25,32)(27,28);(17,19)(18,24)(20,29)(21,30)(22,31)(23,26)(25,28)(27,32);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,16,11)(17,20,22,21)(18,25,26,27)(19,29,31,30)(23,32,24,28);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,11,16,15)(18,26)(23,24)(25,27)(28,32)Polycyclic presentation:
2^9|d^2=h,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,f^b=fh,g^b=gi,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72231256, C2×#6349015, #72306533, … |
| maximal normal subgroup of | #72231256, C2×#6349015, #72306533, … |
| maximal quotient of | #72279203, #72279227, #72279276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256211);