#6687570

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1719 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)(5,6,10,14)(7,15,17,11)(8,12,9,13)(16,20,18,19)(21,22)(23,27)(24,28)(25,30)(26,29)(31,40)(32,47)(33,41)(34,42)(35,43)(36,44)(37,45)(38,48)(39,46)(49,51)(50,52);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23,26,34)(22,27,29,42)(24,35,36,50)(25,32,39,33)(28,43,44,52)(30,47,46,41)(31,38,49,37)(40,48,51,45);(1,2,3,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,24)(22,28)(23,31)(25,38)(26,36)(27,40)(29,44)(30,48)(32,35)(33,50)(34,49)(37,39)(41,52)(42,51)(43,47)(45,46);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(24,36)(28,44)(31,49)(35,50)(37,38)(40,51)(43,52)(45,48);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(22,29)(23,32)(25,39)(27,41)(28,44)(31,35)(33,34)(37,38)(40,52)(42,47)(43,51)(49,50)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6594766);