#128673

order	512
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	D₈
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₈×D₈
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2)(3,4)(5,6,9,16,12,17,26,32,13,18,27,33,11,28,10,15)(7,19,8,14,22,29,24,30,23,35,25,31,21,36,20,34);(2,4)(5,7,10,20,11,21,27,25,13,23,26,24,12,22,9,8)(6,14,16,29,17,30,32,35,18,31,33,36,28,34,15,19);(5,8,9,22,12,24,26,23,13,25,27,21,11,20,10,7)(6,14,16,29,17,30,32,35,18,31,33,36,28,34,15,19)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=gi,e^2=h,f^2=g,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216211, #72216411, #72225396, …
maximal normal subgroup of	#72216211, #72216411, #72225396, …
maximal quotient of	#72225384

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35869);