#147414

order	512
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#466
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

716 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=i,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi,e^c=ei,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54610);