#118095

order	512
nilpotency class	3
isoclinism	isoclinic class of #118091
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

526 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,8,16,10,18,39,54)(3,19,23,56,25,28,50,29)(4,26,30,49,32,42,60,64)(5,33,36,58,6,14,40,55)(7,15,41,52,44,59,37,13)(9,17,46,20,35,57,63,61)(11,22,47,38,31,53,45,21)(12,48,24,34,51,43,62,27);(1,3,5,21,9,24,37,49)(2,11,13,19,17,32,33,34)(4,14,27,18,31,52,28,57)(6,38,35,62,41,64,10,25)(7,42,8,23,36,22,46,51)(12,44,26,39,50,40,53,63)(15,56,20,60,58,43,16,47)(29,61,30,55,48,54,45,59);(1,4,9,31)(2,12,17,50)(3,20,24,16)(5,34,37,19)(6,27,41,28)(7,43,36,56)(8,45,46,30)(10,32,35,11)(13,42,33,22)(14,53,52,26)(15,38,58,64)(18,51,57,23)(21,55,49,59)(25,61,62,54)(29,44,48,40)(39,47,63,60)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,e^b=ei,f^b=fh,d^c=di,e^c=eh,f^c=fhi,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, 25291);