#319626

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
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

756 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)(4,6)(5,8)(9,10,15,18,14,19,13,23)(11,20,21,17,12,16,22,24)(25,26)(27,31)(28,32)(29,33)(30,35)(34,38)(36,40)(37,39);(1,3,4,8)(2,5,6,7)(9,11,14,12)(10,16,19,20)(13,22,15,21)(17,18,24,23)(25,27,34,37,28,35,29,36)(26,30,38,40,32,31,33,39);(9,12,14,11)(10,17,19,24)(13,22,15,21)(16,18,20,23)(25,27,28,35)(26,31,32,30)(29,37,34,36)(33,39,38,40);(2,6)(5,7)(9,13,14,15)(10,18,19,23)(11,21,12,22)(16,17,20,24)(26,32)(30,31)(33,38)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 226822);