Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514148 |
| 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 | |
3342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6)(7,15)(8,17)(9,14)(10,11)(12,18)(13,20)(16,19);(1,2)(3,4)(5,7)(6,8)(9,12)(10,16)(11,18)(13,19)(14,15)(17,20);(2,4)(6,11)(7,16)(12,17)(13,14);(2,4)(5,8)(6,12)(7,13)(9,15)(10,18)(11,17)(14,16)(19,20);(5,8)(6,7)(9,19)(10,18)(11,16)(12,13)(14,17)(15,20)Polycyclic presentation:
2^9|f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,f^b=fi,h^b=hi,h^d=hi,f^e=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72213527, #72219567, #72219569, … |
| maximal normal subgroup of | #72213527, #72219567, #72219569, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421344);