#149245

order	512
nilpotency class	4
isoclinism	isoclinic class of #140362
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1941
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11327
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

503 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)(3,16,21,48)(4,22)(5,27)(6,14)(7,32)(8,13)(9,15)(10,18,37,20)(11,25)(12,30)(17,38,50,60)(19,54,31,47)(23,35,58,62)(24,40)(26,45)(28,42)(29,43)(33,57)(34,41)(36,55,44,53)(39,59)(46,51,64,49)(52,63,56,61);(1,3,5,18,8,19,30,53,9,21,7,20,6,31,29,55)(2,10,12,16,14,36,27,47,15,37,43,48,13,44,32,54)(4,23,24,51,25,56,33,50,26,58,59,49,34,52,28,17)(11,38,39,61,22,46,57,35,41,60,40,63,45,64,42,62);(1,4,6,25,9,26,8,34)(2,11,13,22,15,41,14,45)(3,17,19,49,21,50,31,51)(5,28,29,24,7,33,30,59)(10,35,36,61,37,62,44,63)(12,42,32,39,43,57,27,40)(16,46,47,38,48,64,54,60)(18,52,53,58,20,56,55,23)

Polycyclic presentation:

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

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, 56441);