#220011

order	512
nilpotency class	3
isoclinism	isoclinic class of #164127
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

983 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)(8,16,15,20,17,12,18,19)(21,22)(23,30)(24,28)(25,33)(26,29)(27,31)(32,35)(34,36);(1,2)(3,4)(5,7,10,9)(6,11,13,14)(8,15,17,18)(12,19,16,20);(5,8)(6,12)(7,15)(9,18)(10,17)(11,19)(13,16)(14,20)(21,23,26,32)(22,27,29,36)(24,31,25,34)(28,35,33,30);(2,4)(5,7,10,9)(6,11,13,14)(8,15,17,18)(12,19,16,20)(21,24,26,25)(22,28,29,33)(23,31,32,34)(27,35,36,30)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127207);