#220660

order	512
nilpotency class	3
isoclinism	isoclinic class of #220528
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

787 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127856);