#149161

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=dh,d^2=hi,e^2=h,f^2=hi,g^2=i,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,e^c=ei

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, 56357);