Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3474 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(18,21)(22,24)(25,26)(27,30)(28,32)(29,31);(2,5)(3,8)(9,11,14,12)(10,15,13,16)(19,23)(22,24)(25,27,29,32)(26,30,31,28);(1,3)(2,6)(4,8)(5,7)(9,11,14,12)(10,16,13,15)(17,18)(19,22)(20,21)(23,24)(25,28)(26,27)(29,30)(31,32);(1,3)(2,6)(4,8)(5,7)(9,12,14,11)(10,15,13,16)(17,19)(18,22)(20,23)(21,24)(25,27)(26,28)(29,32)(30,31);(1,2)(3,6)(4,5)(7,8)(9,13,14,10)(11,15,12,16)(17,18)(19,22)(20,21)(23,24)(25,28)(26,27)(29,30)(31,32)Polycyclic presentation:
2^9|b^2=fi,c^2=fh,d^2=fh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232618, #72232624, #72232641, … |
| maximal normal subgroup of | #72232618, #72232624, #72232641, … |
| maximal quotient of | #72251685, #72328353, #72330996 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1953907);