#149357

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#482
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2092
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

436 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)(5,6,12,20,13,21,37,52,14,22,10,18,11,19,9,17)(7,23,28,46,27,43,25,55,29,53,26,15,40,45,38,44)(8,30,35,56,34,48,39,16,36,50,33,49,41,54,32,51)(24,58,61,65,62,67,59,57,63,47,60,68,31,42,64,66);(2,4)(5,7,11,27,14,29,13,40)(6,15,19,44,22,46,21,55)(8,31,41,24,36,62,34,63)(9,26,10,38,37,28,12,25)(16,47,56,42,51,58,49,67)(17,43,18,53,52,45,20,23)(30,57,54,68,50,66,48,65)(32,59,33,60,39,64,35,61);(2,4)(5,8,14,36)(6,16,22,51)(7,24)(9,35,37,33)(10,39,12,32)(11,34,13,41)(15,42)(17,50,52,30)(18,54,20,48)(19,49,21,56)(23,57)(25,60)(26,59)(27,31)(28,64)(29,63)(38,61)(40,62)(43,65)(44,47)(45,68)(46,67)(53,66)(55,58)

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,d^2=hi,e^2=h,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dh,e^b=eh,g^b=gh,h^b=hi,d^c=dh,e^c=ehi,g^c=ghi,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, 56553);