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) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15320 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2325 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,12)(13,15)(14,16)(17,18)(19,24)(20,22)(21,23)(25,26,30,32)(27,31,29,28);(1,3)(2,5)(4,8)(6,7)(9,11,13,16)(10,14,15,12)(17,19)(18,21)(20,24)(22,23)(25,27)(26,28)(29,30)(31,32);(3,8)(5,7)(10,15)(12,14)(18,20)(19,23)(25,28,30,31)(26,27,32,29);(1,3)(2,5)(4,8)(6,7)(9,12,13,14)(10,11,15,16)(17,20,22,18)(19,21,23,24)(25,28)(26,27)(29,32)(30,31);(1,2)(3,7)(4,6)(5,8)(9,11)(10,12)(13,16)(14,15)(17,21)(18,23)(19,20)(22,24)(25,29,30,27)(26,31,32,28)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=hi,d^2=g,e^2=hi,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 | #72300713, #72300748, #72300781 |
| maximal normal subgroup of | #72300713, #72300748, #72300781 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3164428);