#151970

order	512
nilpotency class	5
isoclinism	isoclinic class of #13657
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2177
commutator	C₁₆	C₂×C₄²
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#509
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

280 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59166);