#7167034

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

1728 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)(21,22,28,25)(23,32,36,47)(24,30,39,37)(26,31,27,41)(29,34,40,35)(33,46,49,52)(38,44,42,45)(43,50,51,48);(7,13)(8,16)(10,17)(11,15)(21,23,27,35,28,36,26,34)(22,29,31,32,25,40,41,47)(24,33,38,48,39,49,42,50)(30,43,45,46,37,51,44,52);(2,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,24)(22,30)(23,33)(25,37)(26,42)(27,38)(28,39)(29,43)(31,45)(32,46)(34,50)(35,48)(36,49)(40,51)(41,44)(47,52);(7,13)(10,17)(14,19)(18,20)(21,25,28,22)(23,32,36,47)(24,37,39,30)(26,41,27,31)(29,34,40,35)(33,46,49,52)(38,45,42,44)(43,50,51,48);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,26,28,27)(22,31,25,41)(23,34,36,35)(24,38,39,42)(29,32,40,47)(30,44,37,45)(33,48,49,50)(43,52,51,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,d^2=i,e^2=hi,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, 7074230);