#149733

order	512
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

218 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,20,25,21,26,32,39,22,27,33,40,34,41,45,48,23,28,19,24,29,36,30,37,35,42,31,38,43,46,44,47);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19,21,30,22,31,34,44,23,20,29,32,35,33,43,45)(18,24,26,37,27,38,41,47,28,25,36,39,42,40,46,48)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=fg,e^2=g,g^2=h,h^2=i,b^a=bd,d^a=df,e^b=ef

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, 56929);