Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| 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 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3909 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,22)(21,23)(25,26,28,30)(27,31,32,29);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,13)(14,16)(17,19,21,22)(18,20,23,24)(25,26,28,30)(27,29,32,31);(2,3)(6,7)(10,12)(11,16)(19,22)(20,24)(25,26)(27,29)(28,30)(31,32);(1,3,5,2)(4,6,8,7)(9,12,14,10)(11,13,16,15)(17,20,21,24)(18,22,23,19)(25,26)(27,29)(28,30)(31,32);(1,4)(2,6)(3,7)(5,8)(9,13)(10,16)(11,12)(14,15)(17,18)(19,20)(21,23)(22,24)(25,27)(26,29)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72328073, #72328075, #72328133, … |
| maximal normal subgroup of | #72328073, #72328075, #72328133, … |
| maximal quotient of | #72328544, #72328580 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2646236);