#146022

order	512
nilpotency class	4
isoclinism	isoclinic class of #146008
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2071
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#461
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 237 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=dg,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,h^b=hi,e^c=eg,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, 53218);