#237909

order	512
nilpotency class	3
isoclinism	isoclinic class of #233642
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

868 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,d^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,f^c=fh,g^c=ghi

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, 145105);