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 | |
4731 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,10,13)(7,14,18,20)(8,12,9,19)(11,16,15,17);(1,2)(3,4)(5,7)(6,11)(8,17)(9,16)(10,18)(12,20)(13,15)(14,19);(2,4)(5,6,10,13)(7,15,18,11)(8,12,9,19)(14,16,20,17);(1,2)(3,4)(5,6)(7,11)(8,19)(9,12)(10,13)(14,17)(15,18)(16,20);(2,4)(5,8,10,9)(6,12,13,19)(7,16,18,17)(11,14,15,20)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^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 | #72285851, #72285863, #72286288, … |
| maximal normal subgroup of | #72285851, #72285863, #72286288, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480880);