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 | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2852 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)(9,10,13,11)(12,14,16,15)(17,18,21,23)(19,20,24,22)(25,26)(27,30)(28,29)(31,32);(1,2,5,3)(4,7,8,6)(9,10)(11,13)(12,15)(14,16)(17,19,21,24)(18,22,23,20)(26,28)(30,31);(2,3)(6,7)(10,11)(14,15)(18,23)(20,22)(25,27,29,32)(26,30,28,31);(1,3,5,2)(4,6,8,7)(9,11,13,10)(12,14,16,15)(17,20,21,22)(18,19,23,24)(25,27)(26,31)(28,30)(29,32);(1,4)(2,6)(3,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,20)(21,24)(22,23)(25,28,29,26)(27,31,32,30)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=ghi,d^2=fghi,e^2=ghi,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 | #72329187, #72329200, #72329218 |
| maximal normal subgroup of | #72329187, #72329200, #72329218 |
| maximal quotient of | #72331350 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3238865);