#148561

order	512
nilpotency class	4
isoclinism	isoclinic class of #140211
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1936
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6,11,17,12,19,39,18)(7,20,24,42,25,55,40,54)(8,26,31,48,33,51,32,47)(9,34,35,15,36,16,10,37)(13,23,43,22,45,62,44,63)(14,30,49,29,50,28,52,38)(21,56,59,68,61,41,60,65)(27,66,58,46,57,67,64,53);(5,7,12,25)(6,13,19,45)(8,27,33,57)(9,23,36,62)(10,22,35,63)(11,40,39,24)(14,46,50,53)(15,42,37,54)(16,20,34,55)(17,44,18,43)(21,28,61,30)(26,65,51,68)(29,60,38,59)(31,64,32,58)(41,47,56,48)(49,66,52,67);(2,4)(5,8,9,28,11,32,35,29,12,33,36,30,39,31,10,38)(6,14,15,47,18,49,34,26,19,50,37,48,17,52,16,51)(7,21,22,57,24,60,62,58,25,61,63,27,40,59,23,64)(13,41,20,53,44,65,54,66,45,56,55,46,43,68,42,67)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=h,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,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, 55757);