#152813

order	512
nilpotency class	3
isoclinism	isoclinic class of #11115
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#388
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13121
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

201 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 201 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60009);