#6674263

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42942	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1918 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256919, #72257050, #72257071
maximal normal subgroup of	#72256919, #72257050, #72257071
maximal quotient of	#72264590

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6581459);