#126280

order	512
nilpotency class	4
isoclinism	isoclinic class of #125647
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1619
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#31071	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

380 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33476);