#152850

order	512
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#121
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13159
prime radical	C₂³	#504
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

187 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 187 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323204
maximal normal subgroup of	#72323204

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60046);