#151729

order	512
nilpotency class	5
isoclinism	isoclinic class of #151715
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2246
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	#2246	C₂²
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

217 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 72 points:

(2,5)(3,6)(7,8)(9,10,15,20)(11,21,23,44)(12,27,30,59)(13,31,33,61)(14,34,32,19)(16,35,36,24)(17,39,41,22)(18,42,43,28)(25,55,51,72)(26,56,52,47)(29,60,49,58)(37,67,66,71)(38,54,62,65)(40,50,57,68)(45,64,69,48)(46,63,70,53);(9,11,22,48,13,25,49,65,32,52,68,67,42,63,30,35,15,23,39,64,33,51,29,54,14,26,50,71,28,53,12,24)(10,16,27,46,18,37,57,56,34,62,60,72,61,69,41,44,20,36,59,70,43,66,40,47,19,38,58,55,31,45,17,21);(1,2,6,8,4,7,3,5)(11,23)(16,36)(21,44)(24,35)(25,51)(26,52)(37,66)(38,62)(45,69)(46,70)(47,56)(48,64)(53,63)(54,65)(55,72)(67,71)

Polycyclic presentation:

2^9|a^2=i,b^2=df,c^2=eg,d^2=fh,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,h^a=hi,c^b=ci

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58925);