Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493638 |
| 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 | C2×#41 | C24 |
| prime radical | #63045 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2303 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,8,18,23)(3,11,14,28,12,27,30,32)(4,15,17,26,16,9,19,24)(7,20,22,13,21,31,25,29);(1,3)(2,7)(4,13)(5,14)(6,12)(8,21)(9,11)(10,25)(15,27)(16,29)(17,31)(18,30)(19,20)(22,23)(24,32)(26,28);(3,12)(7,21)(9,15)(13,29)(14,30)(20,31)(22,25)(24,26);(2,8)(5,18)(7,21)(9,15)(11,27)(14,30)(17,19)(20,31)Polycyclic presentation:
2^9|a^2=g,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,c^b=ci,f^b=fhi,g^b=gh,e^c=eh,g^d=gi,f^e=fi,g^e=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72225826, #72232878, #72234103, … |
| maximal normal subgroup of | #72225826, #72232878, #72234103, … |
| maximal quotient of | #72225797, #72226041, #72232771, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400843);