#149341

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	#2073
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)(7,18)(8,23)(9,29)(10,16)(11,35)(12,17)(13,20)(14,26)(15,33)(19,53)(21,51)(22,52)(24,62)(25,46)(27,48)(28,43)(30,39)(31,45)(32,49)(34,65)(36,47)(37,61)(38,57)(40,66)(41,59)(42,56)(44,64)(50,55)(54,68)(58,67)(60,63);(5,7)(6,13)(8,24,28,56)(9,30)(10,21)(11,36)(12,22)(14,42,46,62)(15,47)(16,39)(17,41)(18,29)(19,27,57,37)(20,33)(23,60,43,67)(25,54,26,55)(31,63,64,58)(32,59)(34,66)(35,40)(38,44,53,45)(48,50,61,68)(49,51)(52,65);(5,8,10,27,12,28,11,37)(6,14,35,45,17,46,16,44)(7,19,36,56,22,57,21,24)(9,31,32,26,34,64,33,25)(13,38,39,42,41,53,40,62)(15,48,65,23,49,61,29,43)(18,50,51,67,52,68,47,60)(20,58,66,55,59,63,30,54)

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,d^2=hi,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,d^b=dh,e^b=eh,h^b=hi,d^c=dh,e^c=ehi,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, 56537);