Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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 | #24257 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1677 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,24)(19,20,25,23)(22,26,31,32)(27,30,28,29);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,24)(21,25)(22,27)(26,29)(28,31)(30,32);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(18,24)(19,27)(20,29)(22,31)(23,30)(25,28);(1,4,5,3)(2,6,8,7)(17,20,21,23)(18,25,24,19)(22,30,31,29)(26,28,32,27);(19,27)(20,30)(22,31)(23,29)(25,28)(26,32)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=fi,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72257261, #72257336, #72257471, … |
| maximal normal subgroup of | #72257261, #72257336, #72257471, … |
| maximal quotient of | #72264296 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6488465);