Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667700 |
| 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 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6)(7,11)(8,10)(9,12)(13,14,17,20)(15,18,16,19);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,16)(14,18,20,19);(2,4)(14,19)(15,16)(18,20);(6,10)(9,11);(2,4)(7,12)(9,11)Polycyclic presentation:
2^9|a^2=i,b^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72239595, #72261143, #72261272, … |
| maximal normal subgroup of | #72239595, #72261143, #72261272, … |
| maximal quotient of | #72239669, #72239679, #72305677, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6574914);