#107453

order	512
nilpotency class	3
isoclinism	isoclinic class of #8412
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

773 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:

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

Polycyclic presentation:

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

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, 14649);