#152883

order	512
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²×C₄

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	#13189
prime radical	C₂×#135	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 281 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323203
maximal normal subgroup of	#72323203

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60079);