#191234

order	512
nilpotency class	3
isoclinism	isoclinic class of #191234
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#382
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

608 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,e^2=i,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,g^b=ghi,d^c=dh,g^c=ghi

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, 98430);