#163684

order	512
nilpotency class	3
isoclinism	isoclinic class of #163298
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	#2541	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

849 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 70880);