#152815

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#124
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 ◁ C₁₆ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 204 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60011);