#6572670

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×#3565	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2759 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)(21,22,26,31,27,32,25,30)(23,33,35,42,36,46,40,28)(24,29,37,43,39,45,38,44)(34,47,49,52,50,51,48,41);(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,23,27,36)(22,28,32,42)(24,34,39,50)(25,35,26,40)(29,41,45,52)(30,33,31,46)(37,48,38,49)(43,51,44,47);(2,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,24)(22,29)(23,34)(25,38)(26,37)(27,39)(28,41)(30,44)(31,43)(32,45)(33,47)(35,49)(36,50)(40,48)(42,52)(46,51);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(21,25,27,26)(22,30,32,31)(23,35,36,40)(24,37,39,38)(28,33,42,46)(29,43,45,44)(34,48,50,49)(41,51,52,47);(8,16)(11,18)(14,20)(17,19)(21,25,27,26)(22,30,32,31)(23,35,36,40)(24,38,39,37)(28,33,42,46)(29,44,45,43)(34,49,50,48)(41,47,52,51)

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=h,d^2=i,e^2=i,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, 6479866);