#146768

order	512
nilpotency class	4
isoclinism	isoclinic class of #12750
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(5,6)(7,19,23,46)(8,25)(9,16)(10,33)(11,35)(12,18)(13,21,40,59)(14,27)(15,31)(17,34)(20,53,56,67)(22,41,30,49)(24,39,36,52)(26,50)(28,42)(29,45)(32,48)(37,47)(38,57,66,61)(43,64)(44,63)(51,58,68,55)(54,65,60,62);(1,2,3,4)(5,7)(6,13)(8,20)(9,30)(10,21)(11,36)(12,23)(14,38)(15,39)(16,46)(17,49)(18,40)(19,48)(22,32)(24,34)(25,62)(26,58)(27,54)(28,61)(29,56)(31,59)(33,52)(35,41)(37,57)(42,51)(43,53)(44,66)(45,65)(47,68)(50,67)(55,64)(60,63);(5,8,12,29)(6,14,18,44)(7,20,23,56)(9,27,32,63)(10,26,31,64)(11,37,34,28)(13,38,40,66)(15,45,33,25)(16,47,48,42)(17,50,35,43)(19,51,46,68)(21,58,59,55)(22,60,30,54)(24,61,36,57)(39,65,52,62)(41,53,49,67)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=h,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,d^b=dhi,e^b=ehi,f^b=fi,h^b=hi,d^c=dhi,e^c=eh,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, 53964);