#148976

order	512
nilpotency class	4
isoclinism	isoclinic class of #134690
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#106
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

526 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56172);