#138158

order	512
nilpotency class	4
isoclinism	isoclinic class of #137980
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#472	C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=ghi,e^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45354);