#145130

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

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

normal subgroups only

This group has 337 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:

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

Polycyclic presentation:

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

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, 52326);