#121295

order	512
nilpotency class	3
isoclinism	isoclinic class of #10637
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#397
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1443
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

381 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)(3,4,13,7)(5,10)(6,9)(8,14,15,16)(11,12)(17,18,22,25,23,19,21,27)(20,28,31,26,32,29,30,24);(1,3,11,8,5,13,6,15)(2,7,9,14,10,4,12,16)(17,19,23,18)(20,29,32,28)(21,25,22,27)(24,30,26,31);(1,4,5,7)(2,8)(3,12)(6,14,11,16)(9,13)(10,15)(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=h,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=fi,g^b=ghi,f^c=fhi,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28491);