#7085130

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1271 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)(9,10)(11,13)(12,14)(15,16)(18,24)(21,22)(25,39)(26,40)(27,37)(28,42)(30,36)(32,44)(34,38)(41,45)(43,48)(46,47);(9,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,28,31,25)(20,32,35,40)(21,24,22,37)(26,38,44,34)(29,45,33,48)(30,39,36,42)(41,47,43,46);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19,23,31)(18,25,27,28)(20,33,35,29)(21,36,22,30)(24,39,37,42)(26,43,44,41)(32,45,40,48)(34,46,38,47);(11,16)(13,15)(17,20,23,35)(18,26,27,44)(19,29,31,33)(21,34,22,38)(24,40,37,32)(25,41,28,43)(30,46,36,47)(39,45,42,48);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,22)(18,24,27,37)(19,30,31,36)(20,34,35,38)(25,42,28,39)(26,40,44,32)(29,46,33,47)(41,48,43,45)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,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, 6992326);