#7124195

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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1651 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)(7,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,22,26,30)(23,31,35,41)(24,28,36,43)(25,38,39,29)(27,33,42,34)(32,45,47,51)(37,50,49,44)(40,46,52,48);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,26,35)(22,27,30,42)(24,32,36,47)(25,33,39,34)(28,40,43,52)(29,41,38,31)(37,46,49,48)(44,51,50,45);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24)(22,28)(23,32)(25,37)(26,36)(27,40)(29,44)(30,43)(31,45)(33,46)(34,48)(35,47)(38,50)(39,49)(41,51)(42,52);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(23,33)(25,39)(27,41)(29,38)(31,42)(32,46)(34,35)(37,49)(40,51)(44,50)(45,52)(47,48);(6,12)(10,13)(11,20)(18,19)(24,36)(28,43)(32,47)(37,49)(40,52)(44,50)(45,51)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^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,f^b=fi,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, 7031391);