#162335

order	512
nilpotency class	3
isoclinism	isoclinic class of #162118
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

856 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=gi,d^2=ghi,e^2=h,f^2=i,b^a=be,c^a=cg,d^a=dgh,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, 69531);