#6529847

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500416
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1715 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)(21,22,26,30)(23,31,33,47)(24,28,36,44)(25,38,39,29)(27,40,42,34)(32,46,35,50)(37,52,48,45)(41,51,43,49);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,25,34,26,33,39,40)(22,27,29,31,30,42,38,47)(24,35,37,51,36,32,48,49)(28,43,45,50,44,41,52,46);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24,26,36)(22,28,30,44)(23,32,33,35)(25,37,39,48)(27,41,42,43)(29,45,38,52)(31,46,47,50)(34,49,40,51);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(24,36)(28,44)(32,35)(37,48)(41,43)(45,52)(46,50)(49,51);(6,12)(10,13)(11,20)(18,19)(23,33)(24,36)(27,42)(28,44)(31,47)(34,40)(37,48)(45,52)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6437043);