#106186

order	512
nilpotency class	3
isoclinism	isoclinic class of #106181
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

954 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13382);