#186135

order	512
nilpotency class	3
isoclinism	isoclinic class of #186128
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62781	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2274 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239108
maximal normal subgroup of	#72239108
maximal quotient of	#72239136

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93331);