#195415

order	512
nilpotency class	3
isoclinism	isoclinic class of #18346
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60949	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1245 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fh,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, 102611);