#146758

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	#1862
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

940 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:

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

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,h^a=hi,d^b=dh,e^b=eh,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,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, 53954);