#108663

order	512
nilpotency class	3
isoclinism	isoclinic class of #8760
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1434
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

541 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)(3,5)(4,7)(6,8)(9,10)(11,21,24,16)(12,26,28,34)(13,19)(14,20)(15,18)(17,27,33,29)(22,36)(23,32,25,31)(30,37)(35,38)(39,40);(9,11,14,25,15,24,13,23)(10,16,20,32,18,21,19,31)(12,22,29,38,28,39,27,37)(17,30,34,36,33,35,26,40);(2,5)(3,8)(6,7)(9,12)(10,17,18,33)(11,22)(13,27)(14,29)(15,28)(16,30,21,35)(19,26,20,34)(23,37)(24,39)(25,38)(31,40,32,36)

Polycyclic presentation:

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

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, 15859);