#6687345

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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	#30397	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1777 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:

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

Polycyclic presentation:

2^9|c^2=fgi,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^d=ei

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, 6594541);