#149342

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	#2085
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

428 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,d^2=hi,e^2=hi,f^2=i,g^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, 56538);