#187233

order	512
nilpotency class	3
isoclinism	isoclinic class of #187159
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62748	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2264 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94429);