#118200

order	512
nilpotency class	3
isoclinism	isoclinic class of #118195
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
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

586 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,e^b=ehi,f^b=fh,d^c=di,f^c=fhi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25396);