#148501

order	512
nilpotency class	4
isoclinism	isoclinic class of #13442
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2054
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

643 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=hi,e^2=hi,f^2=h,g^2=i,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,e^c=ei

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, 55697);