#338674

order	512
nilpotency class	3
isoclinism	isoclinic class of #331791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

682 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)(3,5)(4,6)(7,8)(9,10,12,13)(11,14,15,16)(17,18,22,26)(19,23,21,24)(20,28,30,31)(25,29,32,27);(1,3,5,2)(4,7,8,6)(10,13)(11,14)(15,16)(17,19,22,21)(18,23,26,24)(20,29,30,27)(25,28,32,31);(1,4,5,8)(2,6,3,7)(9,11)(10,14)(12,15)(13,16)(18,24)(19,21)(23,26)(25,28)(27,29)(31,32);(2,3)(6,7)(9,10,12,13)(11,14,15,16)(17,20,22,30)(18,25,26,32)(19,27,21,29)(23,31,24,28)

Polycyclic presentation:

2^9|a^2=hi,b^2=ehi,c^2=ei,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,f^a=fh,c^b=cg,d^b=deh,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 245870);