Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,9,11)(7,12,8,10)(14,18)(15,19);(5,6)(7,10)(8,12)(9,11)(14,19)(15,18)(17,20);(1,2,3,4)(13,14)(15,17)(16,18)(19,20);(5,7,9,8)(6,10,11,12);(5,8,9,7)(6,10,11,12)(13,15,16,19)(14,17,18,20)Polycyclic presentation:
2^9|a^2=f,c^2=gi,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72256929, #72257032, #72257039 |
| maximal normal subgroup of | #72256929, #72257032, #72257039 |
| maximal quotient of | #72264699, #72264719 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593927);