#340296

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

659 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cg,d^b=de,e^b=eh,d^c=dh,f^c=fi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 247492);