#323850

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

642 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=eghi,c^2=hi,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 231046);