#339939

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

667 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,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,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, 247135);