#6993961

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	#42796	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2156 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)(17,29)(20,32)(21,33)(22,30)(25,40)(26,36)(27,31)(34,43)(35,41)(37,44)(39,42);(5,6)(7,11)(8,10)(9,12)(13,14,18,22)(15,20,24,35)(16,26,28,33)(17,30,29,19)(21,31,36,27)(23,37,38,43)(25,41,40,32)(34,42,44,39);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15,18,24)(14,20,22,35)(16,23,28,38)(17,25,29,40)(19,32,30,41)(21,34,36,44)(26,37,33,43)(27,39,31,42);(7,12)(9,11)(13,16,18,28)(14,21,22,36)(15,23,24,38)(17,31,29,27)(19,33,30,26)(20,34,35,44)(25,42,40,39)(32,43,41,37);(6,10)(9,11)(15,24)(20,35)(23,38)(25,40)(32,41)(34,44)(37,43)(39,42)

Polycyclic presentation:

2^9|b^2=i,c^2=i,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=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, 6901157);