#220678

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	#61217	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

858 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,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, 127874);