#165100

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	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

700 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)(11,18)(16,23,24,22)(19,20)(25,26,29,34,30,32,28,33)(27,35,36,31,38,40,37,39);(1,2,5,7)(3,6,4,8)(9,10)(11,16)(12,15)(13,21)(14,17)(18,24)(19,23)(20,22)(25,27)(26,31)(28,36)(29,37)(30,38)(32,39)(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)(25,28,30,29)(26,33,32,34)(27,36,38,37)(31,40,39,35)

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=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, 72296);