#164419

order	512
nilpotency class	3
isoclinism	isoclinic class of #164187
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

705 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:

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

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 71615);