Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7504667 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2688 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,7)(3,12)(4,17)(5,21)(8,23)(9,25)(13,22)(15,16)(18,26)(19,20)(27,31)(28,29);(1,2,5,11,6,7,21,10)(3,13,12,8,15,23,16,22)(4,18,17,25,20,9,19,26)(14,29,31,24,30,28,27,32);(1,3)(2,8,7,22)(4,14)(5,16)(6,15)(9,24,18,32)(10,23,11,13)(12,21)(17,27)(19,31)(20,30)(25,28,26,29);(1,4)(2,9)(3,14)(5,19)(6,20)(7,18)(8,24)(10,26)(11,25)(12,27)(13,28)(15,30)(16,31)(17,21)(22,32)(23,29);(3,15)(8,22)(12,16)(13,23)(14,30)(24,32)(27,31)(28,29)Polycyclic presentation:
2^9|b^2=fg,c^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,h^a=hi,c^b=ch,d^b=df,f^b=fi,g^b=gi,e^c=ei,g^c=gi,g^d=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215721, #72216192, #72216217, … |
| maximal normal subgroup of | #72215721, #72216192, #72216217, … |
| maximal quotient of | #72237184, #72250755, C2×#7504667, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411863);