#117990

order	512
nilpotency class	3
isoclinism	isoclinic class of #117987
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

622 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,f^b=fh,d^c=di,e^c=eh,f^c=fhi,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, 25186);