#6358852

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358842
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#61067	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1765 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=eh,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6266048);