#6544248

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2760 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)(21,22,27,31,26,32,25,30)(23,33,35,28,37,43,36,44)(24,38,39,46,41,29,40,45)(34,48,50,52,51,47,49,42);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,26,37)(22,28,32,44)(24,34,41,51)(25,35,27,36)(29,42,38,52)(30,43,31,33)(39,49,40,50)(45,47,46,48);(2,4)(6,10)(11,17)(12,18)(19,20)(21,24)(22,29)(23,34)(25,40)(26,41)(27,39)(28,42)(30,46)(31,45)(32,38)(33,47)(35,50)(36,49)(37,51)(43,48)(44,52);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,25,26,27)(22,30,32,31)(23,35,37,36)(24,39,41,40)(28,43,44,33)(29,45,38,46)(34,49,51,50)(42,47,52,48);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,25,26,27)(22,31,32,30)(23,35,37,36)(24,40,41,39)(28,33,44,43)(29,45,38,46)(34,50,51,49)(42,47,52,48)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6451444);