#136703

order	512
nilpotency class	4
isoclinism	isoclinic class of #13033
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₁₆	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

402 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=g,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,e^b=egi,f^b=fg,g^b=gi,h^b=hi,d^c=dgi,e^c=egi,f^c=fgi,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216421, #72216443, #72216448, …
maximal normal subgroup of	#72216421, #72216443, #72216448, …
maximal quotient of	#72225244, C₂×#136703

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43899);