#6535007

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	C₂×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2734 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)(23,32,35,42)(24,29,36,44)(25,30,26,39)(28,40,43,34)(33,46,47,52)(37,45,38,49)(41,50,51,48);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,27,35)(22,28,31,43)(24,33,36,47)(25,34,26,40)(29,41,44,51)(30,42,39,32)(37,48,38,50)(45,52,49,46);(2,4)(6,10)(11,17)(12,18)(19,20)(21,24)(22,29)(23,33)(25,37)(26,38)(27,36)(28,41)(30,45)(31,44)(32,46)(34,48)(35,47)(39,49)(40,50)(42,52)(43,51);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(24,36)(29,44)(33,47)(37,38)(41,51)(45,49)(46,52)(48,50);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,25,27,26)(22,30,31,39)(23,34,35,40)(24,37,36,38)(28,42,43,32)(29,45,44,49)(33,48,47,50)(41,52,51,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6442203);