Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3903 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,7)(8,10,12,11)(14,18)(15,16)(21,22,25,26)(23,27,28,24);(6,7)(10,11)(13,14,17,18)(15,20,16,19)(22,26)(24,27);(5,6,9,7)(8,11,12,10)(13,15,17,16)(14,19,18,20)(21,23,25,28)(22,27,26,24);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,16,17,15)(14,20,18,19)(21,24)(22,23)(25,27)(26,28);(5,8)(6,10)(7,11)(9,12)(13,14)(15,20)(16,19)(17,18)(21,23)(22,27)(24,26)(25,28)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=f,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232335, #72232345, #72232355, … |
| maximal normal subgroup of | #72232335, #72232345, #72232355, … |
| maximal quotient of | #72278378, #72278779, #72279047, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3947360);