#110868

order	512
nilpotency class	3
isoclinism	isoclinic class of #110864
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

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,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, 18064);