Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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 |
| prime radical | D4×#120 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3594 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)(7,12)(8,10)(9,11)(13,14,17,18)(15,16,19,20);(5,7,9,8)(6,10,11,12)(13,15)(14,16)(17,19)(18,20);(2,4)(14,18)(15,20)(16,19);(5,8,9,7)(6,10,11,12)(13,16,17,20)(14,19,18,15);(14,18)(15,20)(16,19)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=ehi,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72283120, #72305385, #72305388, … |
| maximal normal subgroup of | #72283120, #72305385, #72305388, … |
| maximal quotient of | #72305671, #72305803 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6484000);