#161853

order	512
nilpotency class	3
isoclinism	isoclinic class of #161742
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,6,5,3,7,8)(9,10)(11,18,21,15)(12,16)(13,22)(14,17)(19,24,20,23)(25,26)(27,33)(28,36)(29,31)(30,34)(32,38)(35,40)(37,39);(1,3)(2,5)(4,6)(7,8)(9,11,14,21)(10,15,17,18)(12,19,13,20)(16,23,22,24)(25,27,29,35)(26,30,31,39)(28,37,38,34)(32,40,36,33);(2,3)(6,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(26,31)(30,39)(32,36)(33,40);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(26,31)(30,39)(32,36)(33,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ghi,d^2=ghi,e^2=h,f^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,c^b=ci,e^b=eh,f^b=fgh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 69049);