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×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2837 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)(7,11)(8,10)(9,12)(13,14,17,18)(15,19,20,16);(5,7)(6,9)(8,12)(10,11)(14,18)(15,19)(16,20);(1,2,3,4)(13,15,17,20)(14,19,18,16);(6,10)(9,11);(7,12)(9,11)(13,16)(14,15)(17,19)(18,20)Polycyclic presentation:
2^9|a^2=i,c^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257404, #72257437, #72264241, … |
| maximal normal subgroup of | #72257404, #72257437, #72264241, … |
| maximal quotient of | #72264746, #72315958 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589884);