#254455

order	512
nilpotency class	3
isoclinism	isoclinic class of #254419
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	#24142	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

870 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)(3,11,13,15)(4,9,5,8)(7,12,16,14)(18,23)(20,21)(24,26)(25,32)(27,31)(28,30);(3,12)(4,5)(7,15)(8,9)(11,16)(13,14)(18,24,26,23)(25,32,27,31);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,18)(19,27)(20,24)(21,23)(22,26)(25,29)(28,32)(30,31);(3,13)(7,16)(11,15)(12,14)(17,19)(18,25)(20,30)(21,28)(22,29)(23,31)(24,32)(26,27)

Polycyclic presentation:

2^9|a^2=hi,b^2=e,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,d^c=di,e^c=eh,f^c=fi,g^c=gi,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161651);