#254408

order	512
nilpotency class	3
isoclinism	isoclinic class of #254371
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#120	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

853 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,f^d=fh

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, 161604);