#148495

order	512
nilpotency class	4
isoclinism	isoclinic class of #140148
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1926
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11189
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

639 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=hi,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=dh,e^b=eh,f^b=fhi,h^b=hi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55691);