#119933

order	512
nilpotency class	3
isoclinism	isoclinic class of #10488
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1440
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

541 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27129);