Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(10,14)(11,15)(17,18)(19,23)(20,22)(21,24)(25,26)(27,30)(28,29)(31,32);(1,2)(3,5)(4,7)(6,8)(9,10)(11,12)(13,14)(15,16)(17,19,22,24)(18,21,20,23)(26,29)(30,32);(2,3)(6,7)(10,14)(11,15)(18,20)(19,24)(25,27,28,31)(26,30,29,32);(1,3,5,2)(4,7,8,6)(9,11,13,15)(10,12,14,16)(17,20,22,18)(19,21,24,23)(25,27)(26,30)(28,31)(29,32);(1,4)(2,6)(3,7)(5,8)(9,12)(10,15)(11,14)(13,16)(17,21)(18,19)(20,24)(22,23)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=fh,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
| maximal subgroup of | #72328578, #72328651, #72328667, … |
| maximal normal subgroup of | #72328578, #72328651, #72328667, … |
| maximal quotient of | #72330280, #72330793 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1866809);