#148494

order	512
nilpotency class	4
isoclinism	isoclinic class of #140148
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1926
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11189
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

647 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)(4,23,28,42)(5,13)(6,32)(7,12)(8,34)(9,16)(10,19)(11,26,43,27)(14,31)(15,30)(18,49,51,36)(20,47)(21,37)(22,46)(24,56,57,48)(25,44,60,54)(29,41,59,55)(33,39)(35,45)(38,53)(40,58,63,50)(52,62,61,64);(1,3)(2,10)(4,24)(5,20)(6,33)(7,19)(8,35)(9,22)(11,40)(12,45)(13,17)(14,46)(15,47)(16,39)(18,27)(21,30)(23,49)(25,52)(26,51)(28,57)(29,50)(31,53)(32,37)(34,38)(36,42)(41,64)(43,63)(44,56)(48,54)(55,62)(58,59)(60,61);(1,4,5,25,30,59,6,26,9,28,31,60,8,29,7,27)(2,11,12,41,34,54,14,42,16,43,32,55,15,44,13,23)(3,18,19,50,35,61,53,57,22,51,33,58,21,52,20,24)(10,36,17,48,47,64,37,40,39,49,46,56,38,62,45,63)

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=hi,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=dh,e^b=eh,f^b=fhi,h^b=hi,g^c=gi

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, 55690);