#110324

order	512
nilpotency class	3
isoclinism	isoclinic class of #1913
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂⁴	C₂×C₄²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

751 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17520);