Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| automorphism | Aut(G) order 5242880 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
2701 conjugacy classes of subgroups; only the 460 normal subgroups are shown.
This group has 2701 conjugacy classes of subgroups — too many to draw. Only its 460 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10,13,14)(11,15,16,12)(17,18,20,21)(19,22,24,23)(25,26,28,30)(27,31,32,29);(2,5)(6,7)(10,14)(11,16)(17,18,20,21)(19,23,24,22)(25,26,28,30)(27,29,32,31);(2,5)(6,7)(11,16)(12,15)(18,21)(22,23);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,29)(28,30)(31,32);(1,2)(3,6)(4,5)(7,8)(9,12,13,15)(10,11,14,16)(17,19,20,24)(18,22,21,23)(25,27,28,32)(26,29,30,31)Polycyclic presentation:
2^9|a^2=fghi,b^2=ghi,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
| maximal subgroup of | #72266258, #72266263, #72266558, … |
| maximal normal subgroup of | #72266258, #72266263, #72266558, … |
| maximal quotient of | #72318394 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6247315);