Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573549 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6,8,10)(7,9,11,12)(13,14,16,18)(15,17,20,19);(5,7)(6,9)(8,11)(10,12)(14,17)(15,20)(18,19);(2,4)(7,11)(9,12);(5,7)(6,9)(8,11)(10,12)(14,18)(17,19);(7,11)(9,12)Polycyclic presentation:
2^9|a^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=eh83 conjugacy classes — view the full character table.
| maximal subgroup of | #72237567, #72237570, #72237574, … |
| maximal normal subgroup of | #72237567, #72237570, #72237574, … |
| maximal quotient of | #72249786, #72249816, #72257965, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6498582);