#341490

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, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

512 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=hi,d^2=eh,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, 248686);