Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15310 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2651 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,6,7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,22)(20,23)(21,24)(25,26,29,28)(27,31,32,30);(2,4)(6,8)(10,12)(11,16)(17,19,20,24)(18,21,23,22)(26,28)(30,31);(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,24)(25,26,29,28)(27,30,32,31);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,14)(13,16)(17,18)(19,21)(20,23)(22,24)(25,27)(26,30)(28,31)(29,32);(1,4,5,2)(3,6,7,8)(9,12,13,10)(11,15,16,14)(17,19)(18,22)(20,24)(21,23)(25,28,29,26)(27,30,32,31)Polycyclic presentation:
2^9|a^2=fhi,b^2=ghi,c^2=i,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72266516, #72266517, #72266523, … |
| maximal normal subgroup of | #72266516, #72266517, #72266523, … |
| maximal quotient of | #72334855 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2973717);