#271919

order	512
nilpotency class	3
isoclinism	isoclinic class of #271899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60432	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1065 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 179115);