#147142

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₈×C₂²⋊C₄
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

530 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)(5,6,9,14,10,16,25,36,11,15,26,35,27,37,49,58)(7,17,20,39,21,40,45,61,19,12,28,30,44,31,51,54)(8,13,22,32,23,34,46,56,24,33,47,55,48,57,64,67)(18,38,42,59,43,60,63,68,41,29,50,52,62,53,65,66);(5,7)(6,12,15,17)(8,18)(9,20)(10,21)(11,19)(13,29,33,38)(14,30,35,39)(16,31,37,40)(22,42)(23,43)(24,41)(25,45)(26,28)(27,44)(32,52,55,59)(34,53,57,60)(36,54,58,61)(46,63)(47,50)(48,62)(49,51)(56,66,67,68)(64,65);(2,4)(5,8,9,22,10,23,25,46,11,24,26,47,27,48,49,64)(6,13,14,32,16,34,36,56,15,33,35,55,37,57,58,67)(7,18,20,42,21,43,45,63,19,41,28,50,44,62,51,65)(12,29,30,52,31,53,54,66,17,38,39,59,40,60,61,68)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54338);