#194649

order	512
nilpotency class	3
isoclinism	isoclinic class of #19050
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#3150	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1378 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=h,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fhi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101845);