#145140

order	512
nilpotency class	4
isoclinism	isoclinic class of #145129
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2033
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

329 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 329 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,5)(4,6)(7,8)(9,10)(11,20)(12,16)(13,28)(14,18)(15,19)(17,22)(21,33)(23,34)(24,35)(25,26)(27,32)(29,38)(30,40)(31,36)(37,39);(2,5)(6,7)(9,11,15,25)(10,17,19,34)(12,26,28,20)(13,23,16,22)(14,30,31,33)(18,35,36,39)(21,38,40,27)(24,29,37,32);(2,6)(4,8)(5,7)(9,12,16,10,15,28,13,19)(11,21,23,39,25,40,22,35)(14,18,32,38,31,36,29,27)(17,33,26,24,34,30,20,37)

Polycyclic presentation:

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

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, 52336);