#6529399

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500060
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	#25508	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2777 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)(21,22,25,29,26,30,39,46)(23,31,34,41,33,45,38,27)(24,28,37,44,36,43,51,52)(32,47,48,42,35,49,50,40);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23)(22,27)(24,35)(25,38)(26,33)(28,42)(29,45)(30,41)(31,46)(32,36)(34,39)(37,48)(40,43)(44,47)(49,52)(50,51);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24,26,36)(22,28,30,43)(23,32,33,35)(25,37,39,51)(27,40,41,42)(29,44,46,52)(31,47,45,49)(34,48,38,50);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(24,36)(28,43)(32,35)(37,51)(40,42)(44,52)(47,49)(48,50);(6,12)(10,13)(11,20)(18,19)(23,33)(24,36)(27,41)(28,43)(31,45)(34,38)(37,51)(44,52)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6436595);