Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470122 |
| 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 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2664 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)(7,10)(8,11)(9,12)(14,18)(15,16)(19,20);(1,2,3,4)(7,8)(10,11);(5,7,9,8)(6,10,12,11)(13,14,17,19)(15,18,16,20);(5,8,9,7)(6,11,12,10)(13,15,17,16)(14,18,19,20);(6,12)(10,11)Polycyclic presentation:
2^9|b^2=f,c^2=hi,d^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=dh,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72283050, #72283172, #72283570, … |
| maximal normal subgroup of | #72283050, #72283172, #72283570, … |
| maximal quotient of | #72314318 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394875);