#265327

order	512
nilpotency class	3
isoclinism	isoclinic class of #265327
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#18168	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1055 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172523);