#105613

order	512
nilpotency class	3
isoclinism	isoclinic class of #1828
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁴	C₂×C₄²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

921 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)(11,15,20,28)(12,22,23,16)(13,25)(14,17)(18,26)(19,31)(21,35,34,29)(24,30,37,36)(27,32)(33,38)(39,40);(1,3,5,8,6,7,4,2)(9,11,13,21,14,20,26,34)(10,15,18,29,17,28,25,35)(12,19,24,33,23,32,37,40)(16,27,30,38,22,31,36,39);(9,12,14,23)(10,16,17,22)(11,19,20,32)(13,24,26,37)(15,27,28,31)(18,30,25,36)(21,33,34,40)(29,38,35,39)

Polycyclic presentation:

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

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, 12809);