Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3033 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)(4,6)(5,8)(9,10,13,15)(11,12,16,14)(17,18)(19,20)(21,23)(22,24)(25,26)(27,31)(28,30)(29,32);(1,2,4,6)(3,5,8,7)(9,11)(10,14)(12,15)(13,16)(17,19,21,24)(18,22,23,20)(25,27,30,29)(26,31,28,32);(3,8)(5,7)(10,15)(11,16)(19,24)(20,22)(25,26,30,28)(27,32,29,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,22)(23,24)(25,28,30,26)(27,31,29,32);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,23)(22,24)(25,29,30,27)(26,31,28,32)Polycyclic presentation:
2^9|a^2=fhi,b^2=fg,c^2=fghi,d^2=fghi,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 | #72232597, #72232624, #72232649, … |
| maximal normal subgroup of | #72232597, #72232624, #72232649, … |
| maximal quotient of | #72329048, #72330633 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1890040);