#126779

order	512
nilpotency class	4
isoclinism	isoclinic class of #360
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#102
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	#102
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

572 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gi,e^2=h,f^2=gh,g^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33975);