#146785

order	512
nilpotency class	4
isoclinism	isoclinic class of #12826
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1804
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

935 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)(3,16,20,48,22,49,21,36)(4,23,26,46,28,44,27,40)(5,29,6,11,32,12,34,45)(9,30,37,33,39,19,38,18)(10,24,41,25,43,59,42,58)(17,50,53,35,55,62,54,63)(47,60,64,52,61,51,56,57);(1,3,7,20,8,22,31,21)(2,9,14,38,15,39,13,37)(4,17,27,54,28,55,26,53)(5,30,6,33,32,19,34,18)(10,35,41,62,43,63,42,50)(11,36,29,49,45,48,12,16)(23,56,46,47,44,64,40,61)(24,57,58,51,59,52,25,60);(1,4)(2,10)(3,17)(5,25)(6,24)(7,27)(8,28)(9,35)(11,44)(12,46)(13,42)(14,41)(15,43)(16,47)(18,52)(19,51)(20,54)(21,53)(22,55)(23,45)(26,31)(29,40)(30,60)(32,58)(33,57)(34,59)(36,64)(37,50)(38,62)(39,63)(48,56)(49,61)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,d^b=dhi,e^b=eh,h^b=hi,d^c=dh,e^c=ehi,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, 53981);