#110805

order	512
nilpotency class	3
isoclinism	isoclinic class of #9519
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×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	#1437
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

632 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh

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, 18001);