#6572657

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=hi,d^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,f^b=fi,d^c=di,e^c=ei,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, 6479853);