#108467

order	512
nilpotency class	3
isoclinism	isoclinic class of #106004
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1434
prime radical	C₂⁴	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

509 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15663);