#6677339

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1743 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:

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

Polycyclic presentation:

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

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, 6584535);