#111192

order	512
nilpotency class	3
isoclinism	isoclinic class of #9239
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

669 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)(9,10,13,18,14,17,22,32)(11,15,21,28,20,29,25,33)(12,16,24,30,23,31,26,34)(19,35,36,40,37,27,38,39);(2,5)(3,8)(6,7)(9,11)(10,15,17,29)(12,19)(13,25)(14,20)(16,27,31,35)(18,33,32,28)(21,22)(23,37)(24,38)(26,36)(30,40,34,39);(9,12,22,24,14,23,13,26)(10,16,18,34,17,31,32,30)(11,19,21,38,20,37,25,36)(15,27,33,39,29,35,28,40)

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,e^a=ei,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=di,e^c=ei,f^c=fi

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, 18388);