Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C23×C43 |
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 | |
3168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,22,24,20)(25,26,28,30)(27,29,32,31);(1,3,5,8)(2,6,7,4)(9,11,12,15)(10,13,14,16)(17,19)(18,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32);(3,8)(4,6)(10,14)(13,16)(18,23)(19,24)(25,26,28,30)(27,31,32,29);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,20)(21,23)(22,24)(25,26)(27,29)(28,30)(31,32);(1,4)(2,3)(5,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,20)(18,19)(21,22)(23,24)(25,27)(26,29)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=h,b^2=fgi,c^2=fhi,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 | #72328090, #72328160, #72328183, … |
| maximal normal subgroup of | #72328090, #72328160, #72328183, … |
| maximal quotient of | #72328896, #72330895 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6176984);