#127528

order	512
nilpotency class	4
isoclinism	isoclinic class of #7262
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁵	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

830 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=g,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,d^b=dh,e^b=ei,g^b=gi,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34724);