#161569

order	512
nilpotency class	3
isoclinism	isoclinic class of #161458
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

840 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:

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

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,d^a=dg,e^a=eg,c^b=ci,e^b=eh,f^b=fgh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 68765);