#146777

order	512
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1786
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

960 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)(7,19,23,47)(8,25)(9,16)(10,32)(11,35)(12,18)(13,21,40,57)(14,27)(15,30)(17,34)(20,52,55,67)(22,49,58,41)(24,45,36,39)(26,43)(28,64)(29,46)(31,48)(33,50)(37,42)(38,60,66,56)(44,63)(51,62,65,54)(53,68,59,61);(1,2,3,4)(5,7)(6,13)(8,20)(9,22)(10,21)(11,36)(12,23)(14,38)(15,45)(16,47)(17,49)(18,40)(19,48)(24,34)(25,61)(26,62)(27,59)(28,60)(29,55)(30,57)(31,58)(32,39)(33,54)(35,41)(37,56)(42,65)(43,67)(44,66)(46,68)(50,52)(51,64)(53,63);(5,8)(6,14)(7,20)(9,27)(10,33)(11,37)(12,29)(13,38)(15,46)(16,42)(17,50)(18,44)(19,51)(21,54)(22,59)(23,55)(24,60)(25,32)(26,30)(28,34)(31,63)(35,43)(36,56)(39,61)(40,66)(41,67)(45,68)(47,65)(48,64)(49,52)(53,58)(57,62)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53973);