#7167101

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1721 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22)(23,32)(24,30)(25,26)(27,41)(28,31)(29,34)(33,49)(35,45)(36,52)(37,38)(39,48)(40,43)(42,46)(44,51)(47,50);(7,13)(8,16)(10,17)(11,15)(21,23,27,34,28,35,26,42)(22,29,25,32,31,46,41,45)(24,36,38,50,39,33,43,51)(30,47,40,52,48,44,37,49);(2,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,24,28,39)(22,30,31,48)(23,33,35,36)(25,40,41,37)(26,43,27,38)(29,44,46,47)(32,49,45,52)(34,51,42,50);(7,13)(10,17)(14,19)(18,20)(21,25,28,41)(22,26,31,27)(23,29,35,46)(24,37,39,40)(30,38,48,43)(32,34,45,42)(33,47,36,44)(49,50,52,51);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,26,28,27)(22,25,31,41)(23,34,35,42)(24,38,39,43)(29,45,46,32)(30,37,48,40)(33,50,36,51)(44,49,47,52)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7074297);