#161186

order	512
nilpotency class	3
isoclinism	isoclinic class of #161128
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1144 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=ghi,e^2=h,f^2=i,b^a=be,c^a=ci,d^a=dh,e^a=eg,e^b=eh,f^b=fgh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 68382);