#108866

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#117
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

632 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16062);