#127206

order	512
nilpotency class	4
isoclinism	isoclinic class of #127185
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#342
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

647 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)(3,7)(4,6)(5,8)(9,10)(11,19)(12,16,23,29)(13,25)(14,18)(15,21,27,33)(17,26)(20,31)(22,36)(24,38,37,30)(28,40,39,35)(32,34);(2,6,8,7)(9,11,13,22,14,20,26,34)(10,15,17,28,18,27,25,39)(12,21,24,35,23,33,37,40)(16,19,30,32,29,31,38,36);(1,3,5,4)(2,7,8,6)(11,20)(15,27)(19,31)(21,33)(22,34)(28,39)(32,36)(35,40)

Polycyclic presentation:

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

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, 34402);