Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 2097152 |
| 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 | #15947 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3660 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)(4,6,8,7)(10,11)(12,16)(17,18)(19,20)(21,24)(22,23)(26,29)(27,31);(2,3)(6,7)(9,10,13,11)(12,14,16,15)(17,19)(18,21)(20,23)(22,24)(27,31)(30,32);(1,2,5,3)(4,7,8,6)(9,10,13,11)(12,15,16,14)(17,20,22,21)(18,19,23,24)(25,26)(27,30)(28,29)(31,32);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,15,16,14)(17,21,22,20)(18,19,23,24)(25,27,28,31)(26,30,29,32);(1,4)(2,6)(3,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,20)(21,23)(22,24)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=fghi,c^2=f,d^2=hi,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 | #72258884, #72258885, #72258888, … |
| maximal normal subgroup of | #72258884, #72258885, #72258888, … |
| maximal quotient of | #72318802, #72334903 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2263470);