#146736

order	512
nilpotency class	4
isoclinism	isoclinic class of #12741
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1786
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10788
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

939 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)(3,15)(4,20,22,38)(5,24)(6,13)(7,30)(8,14)(9,17)(10,21,39,56)(11,28)(12,27)(16,48,50,33)(18,44)(19,34)(23,43,32,45)(25,35)(26,41,58,40)(29,46)(31,42)(36,57)(37,54)(47,49,55,63)(51,62,52,61)(53,60,64,59);(1,3,6,18,8,19,7,31)(2,9,13,35,14,37,30,36)(4,16,23,51,22,50,32,52)(5,25,27,54,29,57,28,17)(10,33,40,61,39,48,41,62)(11,42,24,15,12,44,46,34)(20,55,43,59,38,47,45,60)(21,53,26,49,56,64,58,63);(1,4)(2,10)(3,16)(5,26)(6,23)(7,32)(8,22)(9,33)(11,43)(12,45)(13,40)(14,39)(15,47)(17,53)(18,51)(19,50)(20,46)(21,28)(24,38)(25,49)(27,56)(29,58)(30,41)(31,52)(34,55)(35,61)(36,62)(37,48)(42,59)(44,60)(54,64)(57,63)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,d^b=dh,e^b=eh,h^b=hi,d^c=dh,e^c=eh,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53932);