#331280

order	512
nilpotency class	3
isoclinism	isoclinic class of #331276
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

655 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,23)(19,22)(20,24)(25,26,30,33,29,32,38,40)(27,34,35,39,31,28,36,37);(2,5)(6,8)(9,11,12,18,13,19,20,17)(10,14,16,23,15,22,24,21)(25,27,30,35,29,31,38,36)(26,28,33,37,32,34,40,39);(1,3)(2,6)(4,7)(5,8)(11,17)(12,20)(14,21)(16,24)(18,19)(22,23)(25,28,29,34)(26,31,32,27)(30,39,38,37)(33,35,40,36);(3,7)(6,8)(10,15)(14,22)(16,24)(21,23)(27,31)(28,34)(35,36)(37,39)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,c^b=cg,d^b=dh,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238476);