#319755

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

727 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)(9,10,14,19,15,20,13,18)(11,16,21,24,12,23,22,17)(27,35)(31,39)(36,40)(37,38);(1,3,4,8)(2,5,6,7)(9,11,14,21,15,12,13,22)(10,16,19,24,20,23,18,17)(25,26,30,34,29,33,28,32)(27,36,37,39,35,40,38,31);(9,12)(10,17)(11,15)(13,22)(14,21)(16,18)(19,23)(20,24)(25,27,29,35)(26,31,33,39)(28,37,30,38)(32,36,34,40);(2,6)(5,7)(9,13,15,14)(10,18,20,19)(11,21,12,22)(16,24,23,17)(25,28,29,30)(26,32,33,34)(27,37,35,38)(31,36,39,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,c^2=hi,d^2=i,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, 226951);