#161858

order	512
nilpotency class	3
isoclinism	isoclinic class of #161854
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

705 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)(11,17,19,24)(12,16)(13,15)(14,18,22,20)(21,23)(25,26)(27,34)(28,31)(29,33)(30,38)(32,36)(35,39)(37,40);(1,3)(2,5)(4,6)(7,8)(9,11)(10,14)(12,20)(13,19)(15,22)(16,24)(17,23)(18,21)(25,27)(26,32)(28,35)(29,36)(30,39)(31,37)(33,40)(34,38);(2,3)(6,8)(10,15)(14,22)(16,23)(17,24)(25,28,31,26)(27,35,37,32)(29,33,30,38)(34,36,40,39);(25,29,31,30)(26,33,28,38)(27,36,37,39)(32,40,35,34)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 69054);