#190066

order	512
nilpotency class	3
isoclinism	isoclinic class of #190065
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

985 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18)(19,27,30,23)(20,25)(21,24)(22,26)(28,31,29,32);(4,12)(7,14)(10,16)(13,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,32)(26,27);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(18,24,26,25)(23,31,27,32);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,22,21)(18,25,26,24)(19,28,30,29)(23,32,27,31)

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=i,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,g^b=gh,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97262);