#220698

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	#61287	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

807 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)(3,13)(4,14)(5,9)(6,11)(7,10)(8,12)(15,16)(17,18,21,26,22,25,19,23)(20,28,27,32,29,24,30,31);(1,3)(2,4)(5,12)(6,9)(7,14)(8,15)(10,16)(11,13)(17,19,22,21)(18,23,25,26)(20,27,29,30)(24,31,28,32);(1,4,8,5)(2,3,12,15)(6,13,7,16)(9,10,14,11)(17,20)(18,24)(19,27)(21,30)(22,29)(23,31)(25,28)(26,32);(1,5,8,4)(2,9,12,14)(3,10,15,11)(6,13,7,16)(17,19,22,21)(18,23,25,26)(20,27,29,30)(24,31,28,32)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^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, 127894);