Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
3410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(6,7)(10,14)(11,12)(15,16);(1,2)(3,4)(6,7)(9,10,11,16,13,15,12,14);(5,6,8,7)(9,11,13,12)(10,14,15,16);(2,4)(5,7,8,6)(9,11,13,12)(10,14,15,16);(9,10,11,16,13,15,12,14)Polycyclic presentation:
2^9|b^2=fi,c^2=g,d^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257084, #72257098, #72257103, … |
| maximal normal subgroup of | #72257084, #72257098, #72257103, … |
| maximal quotient of | #72264863, #72308274, #72319905 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470463);