#146755

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	#1879
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

948 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)(7,20)(8,26,29,47)(9,31)(10,16)(11,18)(12,35)(13,19)(14,22)(15,28,44,64)(17,34)(21,53,56,67)(23,41)(24,52)(25,49)(27,45,63,46)(30,50,37,43)(32,40)(33,48)(36,39)(38,57,66,58)(42,60)(51,55,68,59)(54,62,61,65);(2,4)(5,7,11,24,13,25,12,36)(6,14,18,40,19,42,35,41)(8,21,30,57,29,56,37,58)(9,32,33,60,34,23,10,22)(15,38,45,53,44,66,46,67)(16,39,31,20,48,52,17,49)(26,62,50,68,47,65,43,51)(27,59,28,54,63,55,64,61);(5,8)(6,15)(7,21)(9,28)(10,27)(11,30)(12,37)(13,29)(14,38)(16,47)(17,50)(18,45)(19,44)(20,51)(22,59)(23,61)(24,57)(25,56)(26,48)(31,43)(32,54)(33,63)(34,64)(35,46)(36,58)(39,65)(40,53)(41,67)(42,66)(49,68)(52,62)(55,60)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=h,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, 53951);