#164968

order	512
nilpotency class	3
isoclinism	isoclinic class of #164187
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

716 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)(3,4)(7,8)(10,15,17,21)(16,22,24,23)(25,26,29,33,30,34,28,32)(27,35,36,31,38,39,37,40);(1,2,5,7)(3,6,4,8)(9,10)(11,16)(12,15)(13,21)(14,17)(18,23)(19,24)(20,22)(25,27)(26,31)(28,36)(29,37)(30,38)(32,39)(33,35)(34,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,20)(13,18)(14,19)(15,22)(17,24)(21,23);(9,12,14,13)(10,15,17,21)(11,18,19,20)(16,23,24,22)(25,28,30,29)(26,32,34,33)(27,36,38,37)(31,39,40,35)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=gh,e^2=h,f^2=i,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh,d^c=dghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 72164);