#153317

order	512
nilpotency class	5
isoclinism	isoclinic class of #153105
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2206
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#2263	C₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 277 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

(6,14)(7,20)(8,10)(9,24)(11,15)(12,23)(16,17)(18,21)(19,34)(22,27)(25,32)(26,33)(28,29)(30,31)(35,36);(1,2)(3,4)(5,6)(7,8)(9,26)(10,14)(11,20)(12,33)(13,19)(15,17)(16,31)(18,23)(21,30)(22,25)(24,36)(27,28)(29,35)(32,34);(2,4)(5,7,8,6,15,14,30,20,12,16,24,18,28,26,32,35,13,22,25,19,29,34,9,27,23,36,31,33,11,21,10,17)

Polycyclic presentation:

2^9|c^2=d,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=eghi,f^a=fgh,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=egh,f^b=fgh,g^b=gh,h^b=hi,e^c=ei

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, 60513);