#109690

order	512
nilpotency class	3
isoclinism	isoclinic class of #9385
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1436
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

524 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,c^b=ch,d^b=di,e^b=ei,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16886);