#170422

order	512
nilpotency class	3
isoclinism	isoclinic class of #170381
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	#2682	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

795 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)(3,5)(4,7)(6,8)(9,10,13,17)(11,15,18,22)(12,14,20,16)(19,21,24,23)(26,31)(28,29)(32,39)(33,37)(34,35)(38,40);(10,14)(12,20)(15,21)(16,17)(19,24)(22,23)(25,26)(27,32)(28,31)(29,37)(30,33)(34,39)(35,38)(36,40);(2,5)(3,8)(6,7)(9,11,13,18)(10,15,17,22)(12,19,20,24)(14,21,16,23)(25,27,30,36)(26,32,33,40)(28,35,29,34)(31,38,37,39);(11,18)(15,22)(19,24)(21,23)(25,28,30,29)(26,31,33,37)(27,34,36,35)(32,39,40,38)

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 77618);