#153346

order	512
nilpotency class	5
isoclinism	isoclinic class of #153083
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2215
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

212 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=de,b^2=dg,d^2=gh,e^2=gi,f^2=g,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,g^a=gh,h^a=hi,c^b=cf,d^c=dg,e^c=eghi,f^c=fgh,g^c=gh,h^c=hi

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, 60542);