#149156

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

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:

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

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=ehi,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fhi,h^b=hi,f^c=fi,g^c=gi

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, 56352);