Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#453 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6)(7,14)(8,12)(9,17)(10,13)(11,15)(16,19)(18,20);(1,2)(3,4)(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,20)(14,17);(2,4)(5,6)(7,11)(8,12)(9,17)(10,13)(14,18)(15,19)(16,20);(1,2)(3,4)(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14);(2,4)Polycyclic presentation:
2^9|d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,e^d=eh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72219652, #72226333, #72226334, … |
| maximal normal subgroup of | #72219652, #72249427, #72249474, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480857);