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 | |
4746 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)(7,13)(8,10)(9,17)(11,16)(12,19)(14,20)(15,18);(1,2)(3,4)(5,7)(6,11)(8,15)(9,18)(10,14)(12,20)(13,17)(16,19);(2,4)(6,8)(7,14)(9,19)(13,20);(1,2)(3,4)(5,8,10,6)(7,15,14,11)(9,17,19,12)(13,16,20,18);(2,4)(6,8)(7,14)(12,17)(16,18)Polycyclic presentation:
2^9|d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,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 | #72285829, #72285929, #72286288, … |
| maximal normal subgroup of | #72285829, #72285929, #72286288, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480885);