Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×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 | |
3297 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,22)(21,23)(25,26)(27,31)(28,29)(30,32);(2,3)(4,8)(9,10,12,14)(11,13,16,15)(17,19)(18,20)(21,22)(23,24)(26,29)(27,32);(2,3)(6,7)(11,16)(13,15)(19,22)(20,24)(25,26,28,29)(27,30,32,31);(1,3,5,2)(4,7,8,6)(9,11,12,16)(10,13,14,15)(17,20,21,24)(18,22,23,19)(25,26)(27,30)(28,29)(31,32);(1,4)(2,6)(3,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,22)(21,23)(25,27,28,32)(26,30,29,31)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=fghi,d^2=ghi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232610, #72232615, #72232640, … |
| maximal normal subgroup of | #72232610, #72232615, #72232640, … |
| maximal quotient of | #72328802, #72328862 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1910504);