#140848

order	512
nilpotency class	4
isoclinism	isoclinic class of #2212
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1982
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

432 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=e,d^2=gi,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,f^b=fg,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, 48044);