#148947

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	#2076
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

444 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)(5,6,9,16,12,15,11,18)(7,19,23,37,24,13,21,39)(8,25,29,14,30,43,27,41)(10,17,31,45,35,44,34,48)(20,51,52,36,54,66,53,49)(22,50,56,46,57,38,32,64)(26,59,60,40,61,58,55,67)(28,47,62,42,63,68,33,65);(5,7,9,21,12,24,11,23)(6,13,16,37,15,19,18,39)(8,26,27,60,30,61,29,55)(10,32,31,57,35,56,34,22)(14,40,25,58,41,67,43,59)(17,46,45,50,44,64,48,38)(20,33,52,63,54,62,53,28)(36,47,66,65,49,68,51,42);(5,8)(6,14,15,41)(7,20)(9,27)(10,33)(11,29)(12,30)(13,36,19,49)(16,25,18,43)(17,47,44,68)(21,52)(22,55)(23,53)(24,54)(26,32)(28,34)(31,63)(35,62)(37,66,39,51)(38,59,50,58)(40,64,67,46)(42,45,65,48)(56,61)(57,60)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,f^b=fh,g^b=gi,h^b=hi,f^c=fh,g^c=gi,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, 56143);