#127949

order	512
nilpotency class	4
isoclinism	isoclinic class of #11704
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

489 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35145);