#274072

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	#3511	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

830 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=hi,d^2=h,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, 181268);