#163884

order	512
nilpotency class	3
isoclinism	isoclinic class of #17603
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

838 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)(12,15)(13,16)(18,23,24,21)(20,22)(26,30)(28,35)(31,37)(32,36)(34,39)(38,40);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,26,29,32)(27,31,33,38)(28,36,35,30)(34,40,39,37);(1,3,5,4)(2,6,8,7)(25,27)(26,31)(28,34)(29,33)(30,37)(32,38)(35,39)(36,40);(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(27,33)(31,38)(34,39)(37,40)

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,e^a=eg,e^b=eh,f^b=fgh,d^c=dh

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, 71080);