Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573523 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #33166 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1951 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,9)(7,10,11,12)(14,18)(15,22)(19,21)(24,25);(6,9)(10,12)(13,14,16,18)(15,23,22,20)(17,19,26,21)(24,28,25,27);(1,2,3,4)(14,19)(15,24)(17,26)(18,21)(22,25)(27,28);(5,7,8,11)(6,10,9,12)(13,15,16,22)(14,20,18,23)(17,24,26,25)(19,27,21,28);(14,21)(15,24)(17,26)(18,19)(20,23)(22,25)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257339, #72257549, #72263838, … |
| maximal normal subgroup of | #72257339, #72257549, #72263838, … |
| maximal quotient of | #72265166, #72280640 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6490896);