#496450

order	512
nilpotency class	4
isoclinism	isoclinic class of #496449
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#21840
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

675 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403646);