#164904

order	512
nilpotency class	3
isoclinism	isoclinic class of #164127
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

715 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 72100);