#165140

order	512
nilpotency class	3
isoclinism	isoclinic class of #164512
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

683 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 72336);