Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493474 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20585 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #62817 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #3484 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,7)(4,11,12,6)(5,8)(9,15,16,13)(10,14);(2,6)(7,13)(8,11)(14,15);(7,14)(13,15);(1,3)(2,7)(4,9)(5,10)(6,13)(8,14)(11,15)(12,16)Polycyclic presentation:
2^9|a^2=g,b^a=be,c^a=cf,e^a=eh,h^a=hi,g^b=gh,d^c=dh,f^d=fi,g^e=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72216329, #72216344, #72216366, … |
| maximal normal subgroup of | #72216329, #72216344, #72216366, … |
| maximal quotient of | #72219306, C2×#493474, #72219393, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400670);