#106454

order	512
nilpotency class	3
isoclinism	isoclinic class of #106434
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

817 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=h,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=dh,f^b=fi,e^c=eh,f^c=fh

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, 13650);