#7149301

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	#30463	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1890 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)(14,19)(15,23)(17,29)(20,34)(21,32)(22,33)(24,38)(26,37)(27,30)(31,36)(35,40)(43,44);(6,9)(7,11)(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)(5,6)(7,10)(8,9)(11,12)(13,15,18,23)(14,20,22,36)(16,24,28,38)(17,25,29,39)(19,31,33,34)(21,35,37,44)(26,40,32,43)(27,41,30,42);(6,9)(10,12)(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);(5,7,8,11)(6,10,9,12)(15,23)(20,36)(24,38)(25,39)(31,34)(35,44)(40,43)(41,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=i,d^2=i,e^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,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, 7056497);