Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
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 | |
3089 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10,14,16)(11,12,13,15)(17,18)(19,23)(20,24)(21,22)(25,26,28,29)(27,31,32,30);(2,4)(3,8)(9,11)(10,15)(12,16)(13,14)(17,18,21,22)(19,20,24,23)(26,29)(30,31);(2,4)(6,7)(10,16)(11,13)(18,22)(20,23)(25,26,28,29)(27,31,32,30);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,16)(14,15)(17,19)(18,23)(20,22)(21,24)(25,26)(27,30)(28,29)(31,32);(1,4,5,2)(3,7,8,6)(9,13,14,11)(10,12,16,15)(17,20,21,23)(18,19,22,24)(25,27)(26,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72327617, #72327660, #72327741, … |
| maximal normal subgroup of | #72327617, #72327660, #72327741, … |
| maximal quotient of | #72329179 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4971041);