#273654

order	512
nilpotency class	3
isoclinism	isoclinic class of #273563
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

849 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=i,d^2=hi,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=ghi,e^d=ei,f^d=fhi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180850);