#337910

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
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×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

679 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)(11,15,16,14)(17,18,20,22)(19,26,27,32)(21,23,29,25)(24,28,31,30);(1,3,5,2)(4,7,8,6)(10,13)(11,14)(15,16)(18,22)(19,27)(23,25)(28,30);(1,4,5,8)(2,6,3,7)(9,11)(10,14)(12,16)(13,15)(18,23)(21,29)(22,25)(24,32)(26,31)(28,30);(2,3)(6,7)(9,10,12,13)(11,14,16,15)(17,19,20,27)(18,24,22,31)(21,30,29,28)(23,32,25,26)

Polycyclic presentation:

2^9|a^2=hi,b^2=ehi,c^2=ehi,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, 245106);