#151758

order	512
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂⁴

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×C₄⋊C₁₆
prime radical	C₂⁴	M₃₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

392 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)(5,6,9,15,10,17,27,39,11,14,28,36,30,37,53,59,12,16,29,38,31,40,54,60,8,24,25,50,26,51,52,66)(7,18,21,42,22,43,47,63,23,33,48,56,49,57,65,67,19,41,44,61,45,62,64,68,20,13,32,34,46,35,55,58);(2,4)(5,7)(6,13,16,33)(8,20)(9,21)(10,22)(11,23)(12,19)(14,18,24,41)(15,34,38,56)(17,35,40,57)(25,32)(26,46)(27,47)(28,48)(29,44)(30,49)(31,45)(36,42,50,61)(37,43,51,62)(39,58,60,67)(52,55)(53,65)(54,64)(59,63,66,68);(7,19)(13,33)(18,41)(20,23)(21,44)(22,45)(32,48)(34,56)(35,57)(42,61)(43,62)(46,49)(47,64)(55,65)(58,67)(63,68)

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=i,e^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, 58954);