#191229

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,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, 98425);