#326017

order	512
nilpotency class	3
isoclinism	isoclinic class of #288899
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×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

610 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)(3,8)(4,6)(5,7)(9,10,15,12)(11,16,13,14)(17,18)(19,22)(20,24)(21,23)(25,29)(26,30)(27,32)(28,31);(1,3,5,4)(2,6,7,8)(9,11,14,10,15,13,16,12)(17,19,25,26,21,28,20,27)(18,22,29,30,23,31,24,32);(1,4,5,3)(2,6,7,8)(9,12,14,11,15,10,16,13)(19,26)(20,25)(22,30)(24,29)(27,28)(31,32);(2,7)(6,8)(9,13,14,12,15,11,16,10)(18,23)(22,31)(24,29)(30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=eghi,c^2=e,d^2=f,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,f^a=fh,c^b=cg,d^c=dh,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, 233213);