Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6361570 |
| 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 | |
1569 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(4,12)(7,15)(8,13)(9,16)(10,14)(17,18,22,24)(19,26,27,32)(20,25,21,29)(23,30,31,28);(1,2)(3,9)(4,13)(5,8)(6,12)(7,10)(11,15)(14,16)(19,27)(23,31)(26,32)(28,30);(17,19)(18,23)(20,28)(21,30)(22,27)(24,31)(25,32)(26,29);(1,3,5,11)(2,7,8,16)(4,14,12,10)(6,15,13,9)(18,24)(23,31)(25,29)(26,32);(2,8)(6,13)(7,16)(9,15)(17,20,22,21)(18,25,24,29)(19,28,27,30)(23,32,31,26)Polycyclic presentation:
2^9|a^2=i,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=eh,f^b=fh,g^c=gi,f^d=fh65 conjugacy classes — view the full character table.
| maximal subgroup of | #72231256, C2×#6361650, #72306657, … |
| maximal normal subgroup of | #72231256, C2×#6361650, #72306657, … |
| maximal quotient of | #72279809, C2×#6361650, #72309507, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6268846);