Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
2227 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,16)(13,15)(17,18,22,26)(19,23,28,31)(20,24,21,25)(27,30,29,32);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,15,16)(19,27)(20,21)(23,30)(24,25)(28,29)(31,32);(9,12,13,11)(10,14,15,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,30)(25,32)(26,31);(1,4,5,3)(2,6,7,8)(17,19)(18,23)(20,27)(21,29)(22,28)(24,30)(25,32)(26,31);(17,20,22,21)(18,24,26,25)(19,27,28,29)(23,30,31,32)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72281542, #72281815, #72282263, … |
| maximal normal subgroup of | #72281542, #72281815, #72282263, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354141);