Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
3164 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)(3,8,4,6)(9,10,13,16)(11,14,12,15)(17,18,22,24)(19,20,21,23)(25,26,28,29)(27,30,31,32);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,16,15)(17,19,22,21)(18,23,24,20)(26,29)(30,32);(3,4)(6,8)(18,24)(19,21)(25,27,28,31)(26,30,29,32);(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,20)(18,19)(21,24)(22,23)(25,26)(27,30)(28,29)(31,32);(1,4,5,3)(2,6,7,8)(9,12,13,11)(10,15,16,14)(17,21,22,19)(18,20,24,23)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,c^2=fh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232640, #72232643, #72232672, … |
| maximal normal subgroup of | #72232640, #72232643, #72232672, … |
| maximal quotient of | #72251706, #72328856, #72328859 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2426895);