Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
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 | |
7091 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,4)(10,14)(11,12)(15,16);(2,4)(5,6)(7,8)(9,10)(11,14)(12,16)(13,15);(6,8)(9,10)(11,14)(12,16)(13,15);(6,8);(9,11,13,12)(10,14,15,16)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72228311, #72237165, #72239231, … |
| maximal normal subgroup of | #72228311, #72237165, #72239231, … |
| maximal quotient of | #72237131, #72237135, #72240227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344714);