#6541157

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1668 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:

(2,3)(5,6)(7,16)(8,13)(9,14)(10,12)(11,15)(17,20)(18,19)(21,22)(23,28)(24,35)(25,30)(26,39)(27,31)(29,36)(32,47)(33,42)(34,43)(37,45)(38,44)(40,46)(41,49)(48,51)(50,52);(1,2,4,3)(5,7,11,10)(6,12,15,16)(8,18,9,17)(13,20,14,19)(21,23,27,34)(22,28,31,43)(24,32,37,50)(25,38,26,33)(29,41,46,51)(30,44,39,42)(35,47,45,52)(36,49,40,48);(1,3,4,2)(5,8,11,9)(6,13,15,14)(7,17,10,18)(12,19,16,20)(21,24)(22,29)(23,32)(25,36)(26,40)(27,37)(28,41)(30,45)(31,46)(33,48)(34,50)(35,39)(38,49)(42,47)(43,51)(44,52);(5,9,11,8)(6,14,15,13)(7,17,10,18)(12,19,16,20)(21,25,27,26)(22,30,31,39)(23,33,34,38)(24,36,37,40)(28,42,43,44)(29,45,46,35)(32,48,50,49)(41,47,51,52);(5,10,11,7)(6,12,15,16)(8,18,9,17)(13,19,14,20)(22,31)(28,43)(29,46)(30,39)(35,45)(41,51)(42,44)(47,52)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6448353);