#274041

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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

812 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=hi,d^2=i,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, 181237);