#222010

order	512
nilpotency class	3
isoclinism	isoclinic class of #164187
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1014 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6,9,14,10,13,7,11)(8,16,15,20,18,12,17,19)(21,22)(23,27)(24,31)(25,34)(26,29)(28,30)(32,36)(33,35);(1,2)(3,4)(5,7,10,9)(6,11,13,14)(8,17,18,15)(12,20,16,19)(21,23,26,25)(22,27,29,34)(24,32,33,30)(28,31,36,35);(5,8,10,18)(6,12,13,16)(7,15,9,17)(11,19,14,20)(21,24)(22,28)(23,30)(25,32)(26,33)(27,35)(29,36)(31,34);(2,4)(21,23,26,25)(22,27,29,34)(24,30,33,32)(28,35,36,31)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 129206);