Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499795 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2545 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,11)(8,10)(9,12)(13,17)(14,19)(15,20)(16,18);(1,2)(3,4)(6,10)(11,19)(12,18)(17,20);(2,4)(5,7,9,15)(6,11,12,20)(8,14,16,13)(10,17,18,19);(5,8)(7,13)(9,16)(11,20)(14,15)(17,19)Polycyclic presentation:
2^9|c^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,h^a=hi,c^b=cg,f^b=fh,d^c=di,e^c=eh,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72209864, #72218077, #72232683, … |
| maximal normal subgroup of | #72218077, #72232683, #72233167, … |
| maximal quotient of | #72209860, #72227781, #72227782, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406993);