#151553

order	512
nilpotency class	5
isoclinism	isoclinic class of #151520
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	C₂×D₄	D₁₆
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ #2169 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

188 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58749);