Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| 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 | #30400 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1729 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)(3,4)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,23,29,32)(20,24,21,25)(27,30,28,31);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,28)(21,27)(22,29)(24,31)(25,30)(26,32);(10,14)(13,16)(19,27)(20,21)(23,30)(24,25)(28,29)(31,32);(2,7)(3,8)(10,14)(13,16);(1,4,5,6)(2,3,7,8)(17,20,22,21)(18,24,26,25)(19,28,29,27)(23,31,32,30)Polycyclic presentation:
2^9|a^2=i,b^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72239589, #72251190, #72251246, … |
| maximal normal subgroup of | #72239589, #72251190, #72251246, … |
| maximal quotient of | #72239747, #72265002 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7412691);