#118719

order	512
nilpotency class	3
isoclinism	isoclinic class of #118710
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
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

565 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)(3,11,12,15)(4,7)(5,9)(6,10)(8,14,16,13)(17,18,22,26,23,19,21,25)(20,24,30,32,29,27,28,31);(2,7)(3,12)(4,5)(8,15)(9,10)(11,16)(17,19,23,18)(20,24,29,27)(21,25,22,26)(28,32,30,31);(1,3,4,13,6,12,5,14)(2,8,9,11,10,16,7,15)(17,20,21,28,23,29,22,30)(18,24,25,31,19,27,26,32)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=d,d^2=i,e^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=eh,g^b=ghi,f^c=fi,g^c=gh

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, 25915);