#148332

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,f^b=fi,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55528);