Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4452694 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
4026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6,9,11)(7,8,12,10)(13,14,17,18)(15,16,20,19)(21,22,25,26)(23,27,28,24);(2,4)(5,7,9,12)(6,10,11,8)(14,18)(15,20)(21,23)(22,24)(25,28)(26,27);(2,4)(6,11)(7,12)(13,15,17,20)(14,19,18,16)(21,24,25,27)(22,23,26,28);(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,27)(26,28);(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,24)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=f,b^2=fhi,c^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72327166, #72327212, #72327257, … |
| maximal normal subgroup of | #72327166, #72327212, #72327257, … |
| maximal quotient of | #72328394, #72332497 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4363451);