#110491

order	512
nilpotency class	3
isoclinism	isoclinic class of #110476
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×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:

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

Polycyclic presentation:

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

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, 17687);