Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
2665 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)(9,10,13,11)(12,14,16,15)(17,18)(19,22)(20,24)(21,23)(27,31)(30,32);(2,3)(4,8)(9,10)(11,13)(12,15)(14,16)(17,19,21,24)(18,22,23,20);(2,3)(6,7)(10,11)(14,15)(17,19)(18,20)(21,24)(22,23)(25,26,29,28)(27,30,31,32);(1,3,5,2)(4,6,8,7)(9,11,13,10)(12,14,16,15)(17,19)(18,20)(21,24)(22,23)(25,27)(26,30)(28,32)(29,31);(1,4,5,8)(2,6,3,7)(9,12,13,16)(10,14,11,15)(17,20)(18,19)(21,22)(23,24)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=fhi,d^2=f,e^2=hi,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3163224);