#6544144

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2713 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)(21,22,27,41,26,31,25,30)(23,32,34,28,36,43,35,44)(24,37,38,46,40,29,39,45)(33,48,51,52,50,47,49,42);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,26,36)(22,28,31,44)(24,33,40,50)(25,34,27,35)(29,42,37,52)(30,43,41,32)(38,49,39,51)(45,47,46,48);(2,4)(6,10)(11,17)(12,18)(19,20)(21,24)(22,29)(23,33)(25,39)(26,40)(27,38)(28,42)(30,46)(31,37)(32,47)(34,51)(35,49)(36,50)(41,45)(43,48)(44,52);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,25,26,27)(22,30,31,41)(23,34,36,35)(24,38,40,39)(28,43,44,32)(29,45,37,46)(33,49,50,51)(42,47,52,48);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6451340);