Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419448 |
| 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:
(1,2,5,12,7,8,6,11)(3,13,16,9,17,25,14,22)(4,18,19,10,20,26,21,23)(15,29,30,27,31,24,28,32);(2,8)(3,14)(5,6)(9,13)(15,28)(16,17)(18,26)(19,21)(22,25)(24,32)(27,29)(30,31);(1,3)(2,9,8,22)(4,15)(5,16)(6,14)(7,17)(10,24,23,29)(11,13,12,25)(18,27,26,32)(19,30)(20,31)(21,28);(1,4)(2,10)(3,15)(5,19)(6,21)(7,20)(8,23)(9,24)(11,18)(12,26)(13,27)(14,28)(16,30)(17,31)(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,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215936, #72216166, #72216203, … |
| maximal normal subgroup of | #72215936, #72216166, #72216203, … |
| maximal quotient of | C2×#419448, #72255856, #72265595 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326644);