Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| automorphism | Aut(G) order 1048576 |
| 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 | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3570 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)(3,6)(4,7)(5,8)(9,10,13,15)(11,12,16,14)(17,18)(19,20)(21,24)(22,23)(26,28)(27,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,24)(21,23)(25,26)(27,31)(28,29)(30,32);(17,20)(18,23)(19,24)(21,22)(25,27)(26,30)(28,31)(29,32);(1,3)(2,6)(4,8)(5,7)(9,12,13,14)(10,11,15,16)(17,20)(18,19)(21,22)(23,24)(25,28,29,26)(27,31,32,30);(1,2)(3,5)(4,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,22)(20,24)(21,23)(25,27)(26,31)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=hi,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232591, #72232606, #72232649, … |
| maximal normal subgroup of | #72232591, #72232606, #72232649, … |
| maximal quotient of | #72328513, #72328684, #72330590 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2561105);