#147671

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

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

1028 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)(14,31)(15,19)(16,22)(17,29)(21,27)(23,34)(24,38)(25,37)(28,35)(30,36)(32,39)(33,40);(9,11,15,25)(10,16,19,33)(12,27,30,39)(13,24,14,23)(17,34,35,38)(18,21,31,32)(20,36,37,26)(22,28,40,29);(2,4)(9,10)(11,21)(12,17)(13,31)(14,18)(15,19)(16,24)(20,27)(22,38)(23,33)(25,32)(26,29)(28,36)(30,35)(34,40)(37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=dh,f^b=fi,h^b=hi,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, 54867);