#336798

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

568 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)(18,23)(19,27)(25,26)(29,30);(1,2)(3,5)(4,6)(7,8)(17,18,21,23)(19,28,27,32)(20,26,22,25)(24,30,31,29);(1,3,4,7)(2,5,6,8)(9,10,13,11)(12,15,16,14)(17,19,21,27)(18,24,23,31)(20,29,22,30)(25,28,26,32);(2,6)(5,8)(9,11,13,10)(12,14,16,15)(17,20,21,22)(18,25,23,26)(19,29,27,30)(24,28,31,32)

Polycyclic presentation:

2^9|a^2=ei,b^2=i,c^2=eh,d^2=hi,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, 243994);