Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2634 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)(3,5)(4,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18,20,21)(19,23,24,22)(26,29)(30,31);(2,3)(4,8)(18,21)(22,23)(25,26,28,29)(27,31,32,30);(2,3)(6,7)(19,24)(22,23)(25,27,28,32)(26,30,29,31);(1,3,5,2)(4,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,19,20,24)(18,22,21,23)(25,27)(26,31)(28,32)(29,30);(1,4,5,8)(2,6,3,7)(9,11,12,15)(10,13,14,16)(17,18)(19,23)(20,21)(22,24)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=h,d^2=fh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72303515, #72303529, #72305958, … |
| maximal normal subgroup of | #72303515, #72303529, #72305958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3140843);