Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6368694 |
| 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 | |
1715 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,7)(4,5)(8,9)(10,15)(11,16)(12,14)(17,18)(19,22)(20,23)(21,25)(24,28)(26,30)(27,29)(31,32);(1,2,6,10)(3,11,13,8)(4,7,5,15)(9,14,16,12)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(1,3)(2,8)(4,14)(5,12)(6,13)(7,9)(10,11)(15,16)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(1,3)(2,9)(4,12)(5,14)(6,13)(7,8)(10,16)(11,15)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(1,4,6,5)(2,7,10,15)(3,12,13,14)(8,16,11,9)(19,26)(22,30)(27,32)(29,31)Polycyclic presentation:
2^9|b^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,c^b=ch,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh65 conjugacy classes — view the full character table.
| maximal subgroup of | #72306531, #72306674, #72306710, … |
| maximal normal subgroup of | #72306531, #72306674, #72306710, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6275896);