#149391

order	512
nilpotency class	4
isoclinism	isoclinic class of #13523
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2092
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11378
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

435 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,d^2=hi,e^2=hi,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=gh,h^b=hi,d^c=dhi,e^c=eh,f^c=fh,g^c=gh,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, 56587);