Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12725 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C24 | C2×C42 |
| Frattini, prime radical, cosocle | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1029 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6)(7,13)(8,16,17,10)(9,12)(11,15)(14,18,20,19);(1,2,3,4)(6,10)(11,18)(12,16)(13,19);(5,7)(6,11,12,13)(8,14)(9,15)(10,18,16,19)(17,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,b^a=bd,c^a=ce,d^a=dh,h^a=hi,f^b=fh,e^c=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72237946, #72237966, #72237993 |
| maximal normal subgroup of | #72237946, #72237966, #72237993 |
| maximal quotient of | #72237932 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53693);