#7124386

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	C₂×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1683 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)(21,22,27,32)(23,33,37,50)(24,29,40,48)(25,31,26,30)(28,42,45,36)(34,44,38,52)(35,43,51,46)(39,49,41,47);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23)(22,28)(24,38)(25,36)(26,42)(27,37)(29,46)(30,33)(31,50)(32,45)(34,40)(35,39)(41,51)(43,48)(44,47)(49,52);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24)(22,29)(23,34)(25,41)(26,39)(27,40)(28,43)(30,49)(31,47)(32,48)(33,44)(35,36)(37,38)(42,51)(45,46)(50,52);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,24)(22,29)(23,35)(25,39)(26,41)(27,40)(28,44)(30,47)(31,49)(32,48)(33,43)(34,36)(37,51)(38,42)(45,52)(46,50);(6,12)(10,13)(11,20)(18,19)(21,25,27,26)(22,30,32,31)(23,36,37,42)(24,39,40,41)(28,33,45,50)(29,47,48,49)(34,51,38,35)(43,52,46,44)

Polycyclic presentation:

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

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, 7031582);