#148147

order	512
nilpotency class	4
isoclinism	isoclinic class of #13310
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1999
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11066
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

673 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fhi,h^b=hi,d^c=di,e^c=eh,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, 55343);