Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(6,9)(7,11);(5,6)(7,10)(8,9)(11,12)(14,18)(15,20)(16,19);(1,2,3,4)(6,9)(10,12)(13,14,19,20,17,15,16,18);(5,7,8,11)(6,10,9,12)(14,15)(18,20);(13,15,19,18,17,14,16,20)Polycyclic presentation:
2^9|c^2=fghi,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72256878, #72256972, #72257050 |
| maximal normal subgroup of | #72256878, #72256972, #72257050 |
| maximal quotient of | #72264553, #72264730 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724467);