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 | |
1609 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)(3,4)(5,13)(6,9)(7,12)(8,11)(10,16)(14,15)(17,18)(19,27)(20,24)(21,28)(22,25)(23,31)(26,32)(29,30);(1,3)(2,7)(4,13)(5,15)(6,11)(8,10)(9,14)(12,16)(17,19)(18,20)(21,29)(22,23)(24,32)(25,30)(26,31)(27,28);(1,3)(2,8)(4,9)(5,12)(6,11)(7,10)(13,14)(15,16)(17,20,22,30)(18,23,25,19)(21,27,32,31)(24,26,29,28);(1,4,6,8)(2,3,9,11)(5,7,16,14)(10,15,13,12)(17,20,22,30)(18,24,25,29)(19,28,23,26)(21,31,32,27);(2,9)(3,11)(10,13)(12,15)(18,25)(20,30)(26,28)(27,31)Polycyclic presentation:
2^9|c^2=i,d^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 | #72306541, #72306549, #72306613, … |
| maximal normal subgroup of | #72306541, #72306549, #72306613, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6276125);