Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493635 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20609 |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #63011 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2663 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)(3,7,11,14)(4,8,12,15)(5,9)(6,10)(13,16);(1,3)(2,6)(4,10)(5,11)(7,8)(9,13)(12,16)(14,15);(4,12)(10,16);(1,3)(2,7)(4,10)(5,11)(6,8)(9,14)(12,16)(13,15)Polycyclic presentation:
2^9|a^2=g,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,f^b=fhi,g^b=gh,e^c=eh,g^d=gi,f^e=fi,g^e=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72216366, #72225822, #72232878, … |
| maximal normal subgroup of | #72216366, #72225822, #72232878, … |
| maximal quotient of | #72225802, #72226081, #72232764, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400831);