#6383945

order	512
nilpotency class	3
isoclinism	isoclinic class of #61058
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#548	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2593 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,c^b=ci,d^b=df,f^b=fi,f^d=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6291141);