#339960

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
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×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

705 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=ehi,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,f^c=fi,f^d=fi

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, 247156);