#209751

order	512
nilpotency class	3
isoclinism	isoclinic class of #18618
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1003 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=ghi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=eh,f^b=fi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116947);