#314083

order	512
nilpotency class	3
isoclinism	isoclinic class of #264283
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

708 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)(3,6)(4,5)(7,8)(9,10,15,12)(11,16,13,14)(17,18,21,25)(19,26,29,23)(20,30,22,24)(27,32,28,31);(2,5)(6,8)(9,11,14,10,15,13,16,12)(17,19,21,29)(18,23,25,26)(20,28,22,27)(24,31,30,32);(1,3,4,7)(2,6,5,8)(9,12,14,11,15,10,16,13)(19,27)(20,22)(23,31)(24,30)(26,32)(28,29);(2,5)(6,8)(9,13,14,12,15,11,16,10)(17,20,21,22)(18,24,25,30)(19,28,29,27)(23,31,26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 221279);