Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23391 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2066 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)(3,6)(4,7)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,23)(21,24)(25,32)(26,30)(27,29)(28,31);(1,3,6,2)(4,8,5,7)(9,11,12,15)(10,13,14,16)(17,19,20,26)(18,22,23,30)(21,25,28,27)(24,32,31,29);(1,4,6,5)(2,7,3,8)(10,14)(13,16)(18,23)(19,25)(21,28)(22,29)(26,27)(30,32);(1,5,6,4)(2,7,3,8)(10,14)(13,16)(18,23)(22,30)(24,31)(29,32);(19,26)(22,30)(25,27)(29,32)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=f,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72281815, #72282126, #72282571, … |
| maximal normal subgroup of | #72281815, #72282126, #72282571, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7413964);