#121010

order	512
nilpotency class	3
isoclinism	isoclinic class of #121008
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

413 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=hi,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=dh,f^b=fhi,g^b=gi,f^c=fi,g^c=ghi

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, 28206);