#319692

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

805 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)(9,10,15,18,14,19,13,20)(11,21,22,24,12,16,23,17)(25,26,30,32,28,33,29,34)(27,35,38,39,37,31,36,40);(1,3,4,8)(2,5,6,7)(9,11,14,12)(10,16,19,21)(13,23,15,22)(17,20,24,18)(25,27,29,36,28,37,30,38)(26,31,34,39,33,35,32,40);(9,12,14,11)(10,17,19,24)(13,23,15,22)(16,20,21,18)(26,32)(27,36)(29,30)(33,34)(37,38)(39,40);(2,6)(5,7)(9,13,14,15)(10,18,19,20)(11,22,12,23)(16,24,21,17)(26,33)(27,37)(32,34)(36,38)

Polycyclic presentation:

2^9|a^2=fh,b^2=eg,c^2=hi,d^2=hi,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, 226888);