#129075

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈²	D₄
commutator	C₄	C₂×C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₄×C₄⋊C₈
prime radical	C₂³	#431
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

320 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=de,d^2=g,f^2=h,h^2=i,b^a=bd,d^b=dg,e^b=eg

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36271);