Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15123 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2593 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,4,7,6)(9,10,12,15)(11,14,16,13)(17,18,21,23)(19,24,22,20)(25,26,30,32)(27,28,31,29);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,15)(13,16)(17,19,21,22)(18,20,23,24)(25,27)(26,29)(28,32)(30,31);(9,11)(10,13)(12,16)(14,15)(17,18)(19,24)(20,22)(21,23)(25,28,30,29)(26,31,32,27);(1,3)(2,4)(5,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,20)(21,23)(22,24)(25,27)(26,28)(29,32)(30,31);(1,4)(2,7)(3,8)(5,6)(9,10)(11,13)(12,15)(14,16)(17,20)(18,22)(19,23)(21,24)(25,29,30,28)(26,31,32,27)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72303423, #72303502, #72303515, … |
| maximal normal subgroup of | #72303423, #72303502, #72303515, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4336076);