#148075

order	512
nilpotency class	4
isoclinism	isoclinic class of #140598
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1961
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:

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

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,f^a=fi,h^a=hi,c^b=cf,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, 55271);