#160605

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

936 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=gh,e^2=h,f^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,e^b=eh,f^b=fgh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 67801);