Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
3337 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,16)(13,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,30)(29,32);(2,6)(3,8)(9,11)(10,14)(12,15)(13,16)(17,19,21,24)(18,20,22,23)(25,26)(27,31)(28,30)(29,32);(3,8)(4,7)(10,15)(11,16)(18,22)(20,23)(25,26,28,30)(27,29,32,31);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,24)(22,23)(25,26)(27,29)(28,30)(31,32);(1,4)(2,3)(5,7)(6,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,23)(22,24)(25,27)(26,29)(28,32)(30,31)Polycyclic presentation:
2^9|b^2=gh,c^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efh,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 | #72300950, #72329071, #72329075, … |
| maximal normal subgroup of | #72300950, #72329071, #72329075, … |
| maximal quotient of | #72330708 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5157791);