Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15259 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2001 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)(3,8)(4,17,19,9)(5,12)(6,10)(7,11)(13,22,29,32)(14,25)(15,24)(16,23)(18,26,30,31)(20,28,21,27);(1,3)(2,8)(4,18,19,30)(5,15)(6,16)(7,14)(9,26,17,31)(10,23)(11,25)(12,24)(13,20,29,21)(22,27,32,28);(1,4)(2,9)(3,13)(5,21)(6,20)(7,19)(8,22)(10,27)(11,17)(12,28)(14,29)(15,30)(16,18)(23,26)(24,31)(25,32);(2,10)(5,6)(8,23)(9,27)(11,12)(15,16)(17,28)(18,30)(20,21)(22,26)(24,25)(31,32)Polycyclic presentation:
2^9|a^2=e,b^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,e^c=ei,g^c=gi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215912, #72216166, #72216958, … |
| maximal normal subgroup of | #72215912, #72216166, #72216958, … |
| maximal quotient of | C2×#417909, #72265716 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325105);