#7167021

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,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, 7074217);