#6687158

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	#42679	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

(2,4)(5,6)(7,15)(8,12)(9,13)(10,14)(11,17)(16,20)(18,19)(21,22,26,30)(23,27,34,41)(24,28,35,42)(25,29,37,44)(31,38,33,40)(32,39,47,51)(36,43,48,52)(45,49,46,50);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23)(22,27)(24,32)(25,31)(26,34)(28,39)(29,38)(30,41)(33,37)(35,47)(36,45)(40,44)(42,51)(43,49)(46,48)(50,52);(1,2,3,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(23,31)(24,35)(25,37)(27,38)(28,42)(29,44)(32,46)(33,34)(39,50)(40,41)(45,47)(49,51);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(21,24,26,35)(22,28,30,42)(23,32,34,47)(25,36,37,48)(27,39,41,51)(29,43,44,52)(31,45,33,46)(38,49,40,50);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(23,31)(25,37)(27,38)(29,44)(32,45)(33,34)(36,48)(39,49)(40,41)(43,52)(46,47)(50,51)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6594354);