#161618

order	512
nilpotency class	3
isoclinism	isoclinic class of #7694
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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	#2856	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

990 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 68814);