#237220

order	512
nilpotency class	3
isoclinism	isoclinic class of #237213
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60805	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1254 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,g^b=gi,d^c=dh,e^c=ei,f^c=fi,g^c=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, 144416);