#336796

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

575 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=i,c^2=eh,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

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, 243992);