#148078

order	512
nilpotency class	4
isoclinism	isoclinic class of #134415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1804
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

656 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)(7,18,22,37)(8,23)(9,14)(10,30)(11,17)(12,20,36,53)(13,25)(15,28)(16,29)(19,47,50,68)(21,35,31,42)(24,56)(26,57)(27,43)(32,38)(33,41)(34,63,66,54)(39,61)(40,60)(44,59)(45,58,64,49)(46,62,67,52)(48,65,51,55);(1,2)(3,4)(5,7,11,22)(6,12)(8,19,26,50)(9,20,29,53)(10,31,33,21)(13,34)(14,18)(15,42)(16,37)(17,36)(23,55)(24,48,27,51)(25,58,60,49)(28,62,59,52)(30,46)(32,63,61,54)(35,44)(38,64)(39,45)(40,66)(41,67)(43,68)(47,56)(57,65);(2,4)(5,8,11,26)(6,13,17,40)(7,19,22,50)(9,27,29,24)(10,32,33,61)(12,34,36,66)(14,39,16,38)(15,43,44,56)(18,45,37,64)(20,51,53,48)(21,54,31,63)(23,41,57,30)(25,59,60,28)(35,47,42,68)(46,55,67,65)(49,62,58,52)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=i,d^2=i,e^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,e^b=ei,g^b=gi,d^c=di,e^c=eh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55274);