#252020

order	512
nilpotency class	3
isoclinism	isoclinic class of #246847
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#45986	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

888 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)(11,14)(12,20)(13,19)(15,17)(16,22)(18,24)(21,23)(25,26)(27,35)(28,34)(29,33)(30,32)(31,37)(36,40)(38,39);(1,3,4,8)(2,5,6,7)(9,11,14,10)(12,21,23,20)(13,19,15,17)(16,24,18,22)(25,27)(26,31)(28,36)(29,37)(30,38)(32,40)(33,35)(34,39);(9,12,14,23)(10,16,11,18)(13,24,15,22)(17,21,19,20)(26,32)(28,30)(31,39)(33,34)(35,40)(36,38);(2,6)(5,7)(9,13,14,15)(10,17,11,19)(12,22,23,24)(16,20,18,21)(25,28,29,30)(26,32,33,34)(27,36,37,38)(31,40,35,39)

Polycyclic presentation:

2^9|b^2=ei,c^2=hi,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dh,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, 159216);