Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573671 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4704 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(6,11)(8,9)(12,19)(13,16)(14,17)(18,20);(1,2)(3,4)(5,6)(7,12)(8,11)(9,16)(10,13)(14,18)(15,20)(17,19);(2,4)(5,7,10,15)(6,12,13,20)(8,14,9,17)(11,18,16,19);(1,2)(3,4)(5,8,10,9)(6,11,13,16)(7,14,15,17)(12,18,20,19);(2,4)Polycyclic presentation:
2^9|c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#3021, #72341471, #72341700, … |
| maximal normal subgroup of | D4×#3021, #72341471, #72341700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480870);