#118548

order	512
nilpotency class	3
isoclinism	isoclinic class of #118543
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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

527 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=hi,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,e^b=eh,f^b=fhi,d^c=di,e^c=ehi,f^c=fh,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, 25744);