Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500740 |
| 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 | |
2747 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,11,14)(7,16,13,18)(8,17,12,9)(10,20,19,15);(1,2)(3,4)(5,7,11,13)(6,12,14,8)(9,19,17,10)(15,18,20,16);(2,4)(5,8)(6,13)(7,14)(9,15)(10,18)(11,12)(16,19)(17,20);(5,9,10,7,11,17,19,13)(6,8,15,18,14,12,20,16);(6,14)(8,12)(15,20)(16,18)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=df,f^b=fi,e^c=ei,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213250, #72213311, #72213426, … |
| maximal normal subgroup of | #72213250, #72213311, #72213426, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409354);