#121289

order	512
nilpotency class	3
isoclinism	isoclinic class of #121286
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1443
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

548 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=ghi,e^c=ehi,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28485);