#122136

order	512
nilpotency class	3
isoclinism	isoclinic class of #122131
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1446
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

605 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fi,d^2=h,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,e^b=eh,d^c=di,e^c=eh,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, 29332);