#513082

order	512
nilpotency class	5
isoclinism	isoclinic class of #512939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

369 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=fg,d^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,c^b=ci,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216261, #72216305, #72216311, …
maximal normal subgroup of	#72216261, #72216305, #72216311, …
maximal quotient of	C₂×#513082, #72321065

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420278);