#105319

order	512
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#95
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄×#95
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=hi,e^2=i,f^2=i,g^2=h,b^a=bd,d^a=dh,f^a=fhi,e^b=ei

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, 12515);