#146180

order	512
nilpotency class	4
isoclinism	isoclinic class of #146171
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2085
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#464
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

187 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 187 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=d,c^2=d,d^2=i,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,e^b=ei,f^b=fh,h^b=hi,e^c=eg,f^c=fh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53376);