#122134

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fh,d^2=h,e^2=i,g^2=i,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, 29330);