#149164

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

422 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)(7,21)(8,27)(9,34)(10,18)(11,16)(12,39)(13,20)(14,23)(15,30)(17,37)(19,35)(22,54)(24,43)(25,50)(26,53)(28,45)(29,49)(31,48)(32,64)(33,46)(36,51)(38,44)(40,42)(41,62)(47,61)(52,55)(56,66)(57,58)(59,63)(60,68)(65,67);(5,7,12,40,13,26,35,25)(6,14,19,51,20,44,39,43)(8,28,32,65,33,60,61,55)(9,24,11,38,10,36,37,23)(15,45,48,52,49,68,59,67)(16,50,34,53,17,42,18,21)(22,30,62,63,57,29,56,31)(27,58,47,41,46,54,64,66);(5,8,11,31,12,32,10,30,13,33,37,63,35,61,9,29)(6,15,34,64,19,48,17,27,20,49,18,47,39,59,16,46)(7,22,36,60,25,56,38,65,26,57,24,28,40,62,23,55)(14,41,42,45,43,58,53,67,44,66,50,68,51,54,21,52)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=dh,d^2=hi,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fhi,h^b=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, 56360);