#220688

order	512
nilpotency class	3
isoclinism	isoclinic class of #220548
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

786 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=de,e^b=eh,f^c=fh,g^c=ghi,e^d=eh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127884);