Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129633 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×D4 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
835 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(9,10)(11,19)(12,22,23,16)(13,17)(14,18)(15,21,24,20);(1,2,4,6)(3,7,8,5)(9,11,14,13)(10,15)(12,20,23,21)(16,17)(18,24)(19,22);(1,3,4,8)(2,5,6,7)(9,12)(10,15,18,24)(11,20)(13,21)(14,23)(16,17,22,19)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=f,e^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=eh,g^b=gh,d^c=dh,g^c=gi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72282800, #72282858, #72282873, … |
| maximal normal subgroup of | #72282800, #72282858, #72282873, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56684);