#6677659

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1744 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6584855);