#7149391

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	#30446	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1906 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,3,4)(7,11)(10,12)(13,14)(15,23)(16,27)(17,30)(18,21)(19,25)(20,28)(22,31)(24,37)(26,36)(29,34)(32,44)(33,40)(35,39)(38,41)(42,43);(6,9)(7,11)(13,15,17,25,18,26,31,40)(14,19,22,23,21,33,30,36)(16,24,29,38,28,39,43,44)(20,32,34,35,27,41,42,37);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,24)(17,29)(18,28)(19,32)(21,27)(22,34)(23,35)(25,38)(26,39)(30,42)(31,43)(33,41)(36,37)(40,44);(6,9)(10,12)(15,25)(17,31)(19,23)(22,30)(24,38)(26,40)(29,43)(32,35)(33,36)(34,42)(37,41)(39,44);(5,7,8,11)(6,10,9,12)(14,21)(16,28)(19,33)(22,30)(23,36)(24,39)(29,43)(38,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,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, 7056587);