#149345

order	512
nilpotency class	4
isoclinism	isoclinic class of #13512
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2085
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11358
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

427 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)(3,17)(4,23,28,57)(5,29)(6,12)(7,15)(8,14)(9,16)(10,19)(11,26,44,31)(13,33)(18,41,52,63)(20,39)(21,55)(22,46)(24,37,53,62)(25,48,34,43)(27,59,36,42)(30,47)(32,45)(35,38)(40,54)(49,51,56,60)(50,61,58,64);(1,3,8,35,9,22,7,21)(2,10,14,39,16,40,15,47)(4,24,27,52,28,53,36,18)(5,30,6,19,33,20,32,54)(11,41,48,37,44,63,43,62)(12,38,13,46,45,55,29,17)(23,56,59,64,57,49,42,61)(25,50,26,51,34,58,31,60);(1,4)(2,11,16,44)(3,18,22,52)(5,31)(6,34)(7,27)(8,36)(9,28)(10,37)(12,23,45,57)(13,42,29,59)(14,43,15,48)(17,49)(19,51,54,60)(20,50,30,58)(21,24,35,53)(25,32)(26,33)(38,64)(39,41)(40,62)(46,56)(47,63)(55,61)

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,d^2=hi,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dh,e^b=eh,h^b=hi,d^c=dhi,e^c=eh,g^c=gi,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, 56541);