#550936

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

2122 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)(11,15)(12,20)(13,14)(16,21)(17,19)(18,23)(22,24)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(6,8)(10,14)(15,17)(16,20)(22,23)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(5,6,7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(1,2,4,3)(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,3,4,2)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fgh,d^2=fgi,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, 458132);