#253026

order	512
nilpotency class	3
isoclinism	isoclinic class of #247527
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

800 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,29,33)(27,31,36,40)(28,32,30,37)(34,38,39,35);(1,3,4,8)(2,5,6,7)(11,20)(12,13)(15,23)(16,17)(19,24)(21,22)(25,27)(26,31)(28,35)(29,36)(30,38)(32,34)(33,40)(37,39);(9,11,14,22)(10,15,18,24)(12,20,13,21)(16,19,17,23)(25,26)(27,34)(28,32)(29,33)(30,37)(31,38)(35,40)(36,39);(2,6)(5,7)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,19,24,23)(25,28,29,30)(26,32,33,37)(27,35,36,38)(31,34,40,39)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160222);