#151890

order	512
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₄	C₂×C₄×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,d^a=di,f^a=fi,c^b=ci

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59086);