#185293

order	512
nilpotency class	3
isoclinism	isoclinic class of #8346
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62814	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2301 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,e^b=ei,f^b=fh,g^b=ghi,e^c=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237274, #72237275, #72239739, …
maximal normal subgroup of	#72237274, #72237275, #72239739, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92489);