#191230

order	512
nilpotency class	3
isoclinism	isoclinic class of #8794
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=e,d^2=hi,e^2=i,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,g^b=ghi,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, 98426);