#274031

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	#61097	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1006 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^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, 181227);