#151989

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#117
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#12662
prime radical	C₂³	#504
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂×M₆₄ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

199 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 199 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225363, #72225380, #72225381, …
maximal normal subgroup of	#72225363, #72225380, #72225381, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59185);