#220419

order	512
nilpotency class	3
isoclinism	isoclinic class of #220264
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	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

799 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=hi,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=de,e^b=eh,g^c=gi,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, 127615);