Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2805 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)(4,8)(9,10)(11,16)(12,13)(14,15)(19,23)(22,24)(25,26,28,30)(27,31,32,29);(2,3)(6,7)(10,15)(11,12)(25,26,28,30)(27,29,32,31);(1,2,5,3)(4,6,8,7)(9,11,14,12)(10,16,15,13)(17,18,21,20)(19,22,23,24)(25,27)(26,29)(28,32)(30,31);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,13,15,16)(17,19,21,23)(18,22,20,24)(25,26)(27,29)(28,30)(31,32);(1,4,5,8)(2,6,3,7)(9,13)(10,12)(11,15)(14,16)(17,20,21,18)(19,24,23,22)(25,27)(26,29)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgh,c^2=ghi,d^2=ghi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3732444);