Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499855 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(6,10)(7,13)(9,15)(11,19)(14,16)(17,20);(1,2,4,3)(5,6)(7,14)(8,10)(9,12)(15,18)(19,20);(5,7)(6,11)(8,14)(9,16)(10,17)(12,19)(13,15)(18,20);(1,3,4,2)(5,6,8,10)(7,11,14,17)(9,12,15,18)(13,20,16,19)Polycyclic presentation:
2^9|b^2=ei,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72271393 |
| maximal normal subgroup of | #72271393 |
| maximal quotient of | #72271414 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407991);