Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #495737 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #21452 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33539 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2875 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1053 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,14,6,13)(3,15,18,10,19,26,25,30)(4,11,21,31,22,32,5,23)(9,24,28,17,29,16,27,20);(1,3)(2,9)(4,20)(5,24)(6,18)(7,25)(8,19)(10,11)(12,27)(13,28)(14,29)(15,31)(16,21)(17,22)(23,26)(30,32);(1,3)(2,10)(4,16)(5,17)(6,25)(7,18)(8,19)(9,11)(12,26)(13,15)(14,30)(20,21)(22,24)(23,27)(28,31)(29,32);(1,4,7,21,8,22,6,5)(2,11,12,31,14,32,13,23)(3,16,18,20,19,24,25,17)(9,26,28,30,29,15,27,10)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,d^b=dh,e^b=ehi,g^b=gi,h^b=hi,f^c=fi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72213937, #72213945, #72214039, … |
| maximal normal subgroup of | #72213937, #72213945, #72214039, … |
| maximal quotient of | #72214781, #72216443, #72224903, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402935);