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) = 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 | |
3107 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)(4,6,8,7)(9,10,13,15)(11,12,16,14)(17,18)(19,23)(20,22)(21,24)(25,26,29,32)(27,28,31,30);(1,2)(3,5)(4,7)(6,8)(9,11,13,16)(10,14,15,12)(17,19,22,21)(18,23,20,24)(25,26)(27,28)(29,32)(30,31);(2,3)(6,7)(10,15)(12,14)(17,18,22,20)(19,24,21,23)(25,27,29,31)(26,30,32,28);(1,3,5,2)(4,6,8,7)(9,12,13,14)(10,11,15,16)(17,20,22,18)(19,23,21,24)(25,27)(26,30)(28,32)(29,31);(1,4)(2,6)(3,7)(5,8)(9,11)(10,12)(13,16)(14,15)(17,21,22,19)(18,23,20,24)(25,28,29,30)(26,31,32,27)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=ghi,d^2=fghi,e^2=ghi,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.
| maximal subgroup of | #72266258, #72266259, #72266263, … |
| maximal normal subgroup of | #72266258, #72266259, #72266263, … |
| maximal quotient of | #72318372 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3238772);