Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419456 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15270 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,21,7,8,6,11)(3,12,15,9,16,25,13,22)(4,17,18,10,20,26,19,23)(14,29,28,32,30,24,31,27);(2,8)(3,13)(5,6)(9,12)(14,28)(15,16)(17,26)(18,19)(22,25)(24,32)(27,29)(30,31);(1,3)(2,9,8,22)(4,14)(5,15)(6,13)(7,16)(10,24,23,29)(11,12,21,25)(17,32,26,27)(18,28)(19,31)(20,30);(1,4)(2,10)(3,14)(5,19)(6,18)(7,20)(8,23)(9,24)(11,26)(12,27)(13,28)(15,31)(16,30)(17,21)(22,29)(25,32)Polycyclic presentation:
2^9|a^2=ef,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215958, #72216203, #72216217, … |
| maximal normal subgroup of | #72215958, #72216203, #72216217, … |
| maximal quotient of | C2×#419456, #72255669, #72314087 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326652);