#273970

order	512
nilpotency class	3
isoclinism	isoclinic class of #273947
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	#3035	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

842 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,d^2=hi,e^2=i,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gh,e^c=ehi,f^c=fhi,g^c=gh,e^d=ei,f^d=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 181166);