#579576

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
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

2142 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)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,26)(27,33)(28,35)(29,37)(30,31)(32,39)(34,36)(38,40);(1,2,4,3)(6,8)(10,11)(15,17)(16,18)(22,23)(26,31)(28,33)(32,37)(36,40);(5,6)(7,8)(9,11,14,10)(12,17,20,15)(13,16,21,18)(19,22,24,23)(25,27)(26,28)(29,38)(30,35)(31,33)(32,40)(34,39)(36,37);(1,3,4,2)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,38)(39,40);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29,30,39)(26,32,31,37)(27,34,35,38)(28,36,33,40)

Polycyclic presentation:

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

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, 486772);