#186200

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	#62805	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2237 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=hi,e^2=i,f^2=h,g^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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93396);