#152983

order	512
nilpotency class	5
isoclinism	isoclinic class of #151127
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2177
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

267 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(2,4)(5,6)(7,19,22,46)(8,24,27,52)(9,29)(10,34)(11,38)(12,18)(13,21,40,30)(14,26,42,66)(15,20)(16,36)(17,35)(23,60,59,62)(25,63)(28,51,57,67)(31,50)(32,43)(33,39)(37,47)(41,54,61,64)(44,55)(45,53,68,58)(48,65)(49,56);(1,2,3,4)(5,7,20,53,37,59,32,66,35,27,65,54,36,28,9,30,12,22,56,58,10,23,55,26,11,8,25,64,31,57,33,21)(6,13,29,67,50,61,48,24,38,14,43,60,34,68,15,46,18,40,39,51,16,41,63,52,17,42,44,62,47,45,49,19);(5,8,25,53,10,28,9,26,35,7,20,54,31,23,55,21,12,27,65,58,37,57,33,66,11,22,56,64,36,59,32,30)(6,14,44,51,16,45,15,24,38,40,29,62,47,61,63,46,18,42,43,67,50,68,49,52,17,13,39,60,34,41,48,19)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=egh,d^2=ghi,e^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=ei,f^b=fi,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60179);