#550147

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2190 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=ghi,d^2=fg,e^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df

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, 457343);