Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563341 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2852 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,13,9)(7,15,20,17)(8,11,10,12)(14,19,16,18);(1,2)(3,4)(5,7,11,18,13,20,12,19)(6,14,8,15,9,16,10,17);(5,8,11,9,13,10,12,6)(7,16,19,15,20,14,18,17);(2,4)(5,9,11,10,13,6,12,8)(7,15,19,14,20,17,18,16);(5,10,11,6,13,8,12,9)(7,17,18,14,20,15,19,16)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=g,e^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213663, #72219569, #72219574, … |
| maximal normal subgroup of | #72213663, #72219569, #72219574, … |
| maximal quotient of | #72226898, #72226905, #72226937 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470538);