Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493411 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1781 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #62842 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
3445 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)(3,12)(4,16,18,9)(5,10)(6,11)(7,13,22,27)(8,17,24,19)(14,26)(15,21)(20,25)(23,28,32,30)(29,31);(1,3)(2,7)(4,13)(5,15)(6,14)(8,21)(9,12)(10,23)(11,22)(16,26)(17,28)(18,27)(19,30)(20,29)(24,31)(25,32);(1,4)(2,8)(3,13)(5,19)(6,18)(7,21)(9,10)(11,24)(12,23)(14,27)(15,30)(16,25)(17,20)(22,31)(26,32)(28,29);(3,14)(7,22)(12,26)(13,27)(15,29)(21,31)(23,32)(28,30)Polycyclic presentation:
2^9|a^2=g,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,g^b=gh,f^c=fi,g^c=gi,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72234206, #72234211, #72234212, … |
| maximal normal subgroup of | #72234206, #72234211, #72234212, … |
| maximal quotient of | #72238620, #72312249, C2×#493411, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400607);