#324620

order	512
nilpotency class	3
isoclinism	isoclinic class of #278987
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

620 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)(4,5)(6,8)(9,10,13,16)(11,18,20,15)(12,22,17,14)(19,21,23,24)(25,26)(27,30)(28,32)(29,31)(33,37)(34,38)(35,40)(36,39);(2,5)(6,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,18)(17,23)(22,24)(25,27,33,34,29,36,28,35)(26,30,37,38,31,39,32,40);(9,12,14,10,13,17,22,16)(11,19,21,15,20,23,24,18)(27,34)(28,33)(30,38)(32,37)(35,36)(39,40);(1,3,4,8)(2,6,5,7)(10,16)(11,20)(12,17)(21,24)(26,31)(30,39)(32,37)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=f,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cg,d^b=dh,d^c=dh,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, 231816);