#148779

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	#2033
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

472 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)(7,20)(8,26)(9,18)(10,33)(11,15)(12,19)(13,22)(14,29)(16,36)(17,32)(21,54)(23,59)(24,41)(25,51)(27,53)(28,48)(30,45)(31,62)(34,47)(35,44)(37,46)(38,42)(39,50)(40,60)(43,56)(49,58)(52,64)(55,65)(57,68)(61,67)(63,66);(5,7,9,22,12,25,11,38)(6,13,15,20,19,42,18,51)(8,27,28,60,31,63,39,61)(10,34,32,23,37,58,36,24)(14,43,44,68,45,65,52,54)(16,47,46,41,17,49,33,59)(21,35,55,29,57,64,56,30)(26,40,50,53,62,67,48,66);(5,8,9,28,12,31,11,39)(6,14,15,44,19,45,18,52)(7,21,22,55,25,57,38,56)(10,35,32,29,37,64,36,30)(13,40,20,53,42,67,51,66)(16,48,46,26,17,50,33,62)(23,60,58,63,24,61,34,27)(41,54,49,43,59,68,47,65)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,e^c=ehi,f^c=fh,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, 55975);