Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6451734 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2404 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3869 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,8)(5,6)(9,24)(10,23)(11,20)(12,22)(14,21)(16,29)(17,18)(25,32)(26,31)(27,30);(1,2)(3,12)(4,16)(5,8)(6,20)(7,11)(9,15)(10,19)(13,26)(14,24)(17,29)(18,23)(21,22)(25,28)(27,32)(30,31);(1,3)(2,9,11,12)(4,13)(5,14)(6,21)(7,15)(8,22,20,24)(10,25,16,26)(17,27)(18,30)(19,28)(23,31,29,32);(1,4)(2,10,11,16)(3,13)(5,17)(6,18)(7,19)(8,23,20,29)(9,25,12,26)(14,27)(15,28)(21,30)(22,31,24,32);(1,5,7,6)(2,8,11,20)(3,14,15,21)(4,17,19,18)(9,22,12,24)(10,23,16,29)(13,27,28,30)(25,31,26,32)Polycyclic presentation:
2^9|c^2=g,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72233772, #72233773, #72233777, … |
| maximal normal subgroup of | #72233772, #72233773, #72233777, … |
| maximal quotient of | #72317836, #72318269, #72318575, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6358932);