#148119

order	512
nilpotency class	4
isoclinism	isoclinic class of #13318
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

681 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,d^c=di,e^c=eh,f^c=fhi,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, 55315);