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 | #34964 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1709 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(10,13)(11,15)(17,18)(19,22)(20,23)(21,24)(25,32)(26,30)(27,29)(28,31);(1,3,6,2)(4,8,5,7)(9,10,12,13)(11,14,15,16)(17,19,20,26)(18,22,23,30)(21,25,28,27)(24,32,31,29);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(18,23)(19,25)(21,28)(22,29)(26,27)(30,32);(2,3)(7,8)(10,13)(14,16)(18,23)(22,30)(24,31)(29,32);(1,5,6,4)(2,8,3,7)(19,26)(22,30)(25,27)(29,32)Polycyclic presentation:
2^9|b^2=fi,c^2=f,e^2=fgi,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 | #72251184, #72251197, #72256971 |
| maximal normal subgroup of | #72251184, #72251197, #72256971 |
| maximal quotient of | #72265104 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7420471);