Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,11)(7,8,12,10)(13,14)(15,20)(16,18)(17,19)(21,22)(23,24)(25,26)(27,28);(2,4)(5,7)(6,10)(8,11)(9,12)(13,15,17,18)(14,16,19,20)(22,26)(23,28);(1,2)(3,4)(21,23)(22,27)(24,26)(25,28);(5,8,9,10)(6,7,11,12)(13,14)(15,16)(17,19)(18,20)(21,23)(22,24)(25,28)(26,27);(5,7)(6,8)(9,12)(10,11)(13,16)(14,18)(15,19)(17,20)(21,24)(22,28)(23,26)(25,27)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,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 | #72232867, #72232883, #72232963, … |
| maximal normal subgroup of | #72232867, #72232883, #72232963, … |
| maximal quotient of | #72323623, #72330319, #72330482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2565161);