#127422

order	512
nilpotency class	4
isoclinism	isoclinic class of #127265
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

413 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=ghi,e^2=hi,f^2=ghi,g^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,c^b=ci,d^b=di,e^b=eh

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, 34618);