#184340

order	512
nilpotency class	3
isoclinism	isoclinic class of #3186
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄²×D₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

670 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91536);