#237915

order	512
nilpotency class	3
isoclinism	isoclinic class of #233734
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

782 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)(11,17)(12,20)(13,15)(14,19)(16,21)(18,24)(22,23)(28,36)(30,39)(32,37)(35,38);(1,3,4,8)(2,5,6,7)(12,21)(14,23)(16,20)(19,22)(25,26)(27,33)(28,37)(29,31)(30,35)(32,36)(34,40)(38,39);(9,11,13,18)(10,14,15,23)(12,22,21,19)(16,24,20,17)(25,27)(26,30)(28,38)(29,34)(31,39)(32,40)(33,37)(35,36);(10,15)(11,18)(16,20)(19,22)(27,34)(30,39)(33,40)(35,38)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=h,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,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145111);