#149343

order	512
nilpotency class	4
isoclinism	isoclinic class of #13490
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2073
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)(3,15,21,50)(4,23)(5,30)(6,11)(7,34)(8,14)(9,18,38,32)(10,26)(12,31)(13,16)(17,51,54,40)(19,37,56,39)(20,36,22,45)(24,59,58,35)(25,43)(27,41)(28,48)(29,46)(33,47)(42,61)(44,60)(49,55,64,53)(52,62,57,63);(1,3,16,22,8,21,7,20)(2,9,13,37,14,38,34,39)(4,24,27,54,29,58,28,17)(5,19,33,18,31,56,6,32)(10,40,43,35,44,51,42,59)(11,45,30,50,47,36,12,15)(23,49,41,62,46,64,48,63)(25,52,60,55,61,57,26,53);(1,4,8,29)(2,10)(3,17)(5,26,31,60)(6,25,33,61)(7,27,16,28)(9,35,38,59)(11,46)(12,48)(13,42)(14,44)(15,49,50,64)(18,55)(19,57)(20,24)(21,54)(22,58)(23,47)(30,41)(32,53)(34,43)(36,62,45,63)(37,40,39,51)(52,56)

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,d^2=h,e^2=hi,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=dhi,e^b=ehi,g^b=gi,h^b=hi,d^c=dh,e^c=ehi,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, 56539);