#7160908

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45316	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1558 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,4,5)(3,6,7,8)(11,15)(14,16)(17,18)(19,27)(20,24)(21,35)(22,25)(23,31)(26,36)(28,43)(29,39)(30,38)(32,46)(33,41)(34,48)(37,44)(40,45)(42,47);(10,13)(11,15)(17,19)(18,23)(20,32)(21,29)(22,30)(24,40)(25,39)(26,38)(27,35)(28,33)(31,36)(34,44)(37,41)(42,43)(45,47)(46,48);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,33)(18,24,25,41)(19,28,30,32)(21,34,36,47)(23,37,39,40)(26,42,35,48)(27,43,38,46)(29,45,31,44);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,29)(20,33)(21,36)(23,38)(24,41)(26,35)(27,39)(28,44)(30,31)(32,45)(37,43)(40,46);(9,11,12,15)(10,14,13,16)(18,25)(19,30)(20,33)(26,35)(29,31)(34,47)(37,40)(43,46)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=g,e^2=hi,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, 7068104);