#148588

order	512
nilpotency class	4
isoclinism	isoclinic class of #137786
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1899
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂	#11197
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #3428 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

633 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,f^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,f^b=fi,g^b=gi,f^c=fi,g^c=gh,g^e=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, 55784);