#7003305

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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1875 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)(5,6)(7,9)(8,10)(11,12)(14,19)(15,23)(17,29)(20,34)(21,32)(22,33)(24,38)(26,37)(27,30)(31,36)(35,40)(43,44);(5,6)(7,11)(8,10)(9,12)(13,14,17,19,18,22,29,33)(15,20,25,31,23,36,39,34)(16,26,27,37,28,32,30,21)(24,40,41,44,38,43,42,35);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,24)(17,25)(18,23)(19,31)(21,35)(22,36)(26,40)(27,41)(28,38)(29,39)(30,42)(32,43)(33,34)(37,44);(7,12)(9,11)(13,16,18,28)(14,21,22,37)(15,24,23,38)(17,30,29,27)(19,32,33,26)(20,35,36,44)(25,42,39,41)(31,43,34,40);(6,10)(9,11)(14,22)(15,23)(19,33)(21,37)(24,38)(25,39)(26,32)(41,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,d^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,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, 6910501);