#111146

order	512
nilpotency class	3
isoclinism	isoclinic class of #9183
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

660 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=di

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, 18342);