#153264

order	512
nilpotency class	5
isoclinism	isoclinic class of #153078
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#2246	C₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

365 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,4)(6,7)(9,10)(11,15)(12,24)(13,27)(14,19)(16,20)(17,21)(18,25)(22,34)(23,30)(26,40)(28,36)(29,35)(31,39)(32,37)(33,38);(5,6)(7,8)(9,10)(11,15)(12,24)(13,27)(14,19)(16,20)(17,21)(18,25)(22,34)(23,30)(26,40)(28,36)(29,35)(31,39)(32,37)(33,38);(2,4)(5,7,8,6)(10,15)(11,20)(12,21)(13,25)(16,17)(18,34)(19,30)(22,39)(23,37)(24,28)(26,38)(27,29)(31,40)(32,33)(35,36)

Polycyclic presentation:

2^9|c^2=d,e^2=gh,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fg,g^a=gh,h^a=hi,c^b=cf,e^b=eg,f^b=fg,g^b=gh,h^b=hi,e^c=eg,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321065, #72322860, #72322876
maximal normal subgroup of	#72321065, #72322860, #72322876
maximal quotient of	#72321178, #72322045

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60460);