#119369

order	512
nilpotency class	3
isoclinism	isoclinic class of #119368
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

609 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26565);