#148922

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1832
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

458 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,d^b=di,g^b=gi,e^c=ei,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56118);