#186987

order	512
nilpotency class	3
isoclinism	isoclinic class of #186985
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62835	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ 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)(11,19,22,15)(12,23,25,16)(13,26)(14,18)(17,28)(20,29)(21,30,27,32)(24,31,38,36)(33,35)(34,40)(37,39);(1,2)(3,7)(4,6)(5,8)(9,11,14,22)(10,15,18,19)(12,20,25,35)(13,27,28,21)(16,29,23,33)(17,32,26,30)(24,37,38,34)(31,40,36,39);(3,8)(5,7)(9,12)(10,16)(11,20)(13,24)(14,25)(15,29)(17,31)(18,23)(19,33)(21,34)(22,35)(26,36)(27,37)(28,38)(30,39)(32,40);(9,13,14,28)(10,17,18,26)(11,21,22,27)(12,24,25,38)(15,30,19,32)(16,31,23,36)(20,34,35,37)(29,39,33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239886
maximal normal subgroup of	#72239886
maximal quotient of	#72239896

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94183);