#127346

order	512
nilpotency class	4
isoclinism	isoclinic class of #127249
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁵	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=ghi,f^2=ghi,g^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,c^b=ci,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, 34542);