#160710

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1016 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=i,e^2=h,f^2=i,b^a=be,c^a=ci,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, 67906);