#151900

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₃₂	D₄
commutator	C₄	C₂×C₄×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	D₄×C₁₆
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₂ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

276 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)(5,6,9,16,10,18,26,41,11,15,27,38,29,39,51,61,12,17,28,40,30,42,52,62,8,13,24,32,25,33,50,55)(7,14,20,35,21,37,45,59,22,34,46,56,48,57,64,67,23,36,47,58,49,60,65,68,19,31,43,53,44,54,63,66);(2,4)(6,13,17,15)(14,31,36,34)(16,32,40,38)(18,33,42,39)(35,53,58,56)(37,54,60,57)(41,55,62,61)(59,66,68,67);(5,7,9,20,10,21,26,45,11,22,27,46,29,48,51,64,12,23,28,47,30,49,52,65,8,19,24,43,25,44,50,63)(6,14,16,35,18,37,41,59,15,34,38,56,39,57,61,67,17,36,40,58,42,60,62,68,13,31,32,53,33,54,55,66)

Polycyclic presentation:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59096);