#161653

order	512
nilpotency class	3
isoclinism	isoclinic class of #7694
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

828 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 68849);