Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498379 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23029 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33618 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1765 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,14,9,15,28,23,10,16,6,26,7,21,29,32)(3,17,20,30,19,25,13,31,4,5,18,24,22,12,27,11);(1,3)(2,11,16,31)(4,10)(5,23,17,32)(6,27)(7,19)(8,13)(9,22)(12,14,25,26)(15,24,21,30)(18,29)(20,28);(1,4)(2,12)(3,10)(5,21)(6,20)(7,19)(8,18)(9,22)(11,14)(13,29)(15,17)(16,25)(23,30)(24,32)(26,31)(27,28);(1,5,6,24,9,12,29,11,10,17,8,30,7,25,28,31)(2,13,14,3,15,18,23,19,16,27,26,4,21,20,32,22)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=e,e^2=h,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,d^b=de,e^b=eh,g^b=gh,h^b=hi,f^c=fi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72214039, #72285481, #72285521, … |
| maximal normal subgroup of | #72214039, #72285481, #72285521, … |
| maximal quotient of | #72216456, #72227172, #72227173, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405575);