#138931

order	512
nilpotency class	4
isoclinism	isoclinic class of #138850
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#33520	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

533 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 533 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,5,7,8)(4,6)(9,10,15,21,16,22,13,20,17,23,31,39,30,38,14,32)(11,19,25,18,26,35,12,28,27,34,29,36,33,40,24,37);(1,3,4,7)(2,5)(6,8)(9,11,16,33,17,27,30,26)(10,18,22,37,23,36,38,28)(12,13,25,31,24,14,29,15)(19,32,40,21,34,20,35,39);(9,12)(10,19)(11,13)(14,27)(15,26)(16,25)(17,24)(18,32)(20,36)(21,37)(22,40)(23,34)(28,39)(29,30)(31,33)(35,38)

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,d^2=ghi,e^2=ghi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=eghi,f^b=fh,h^b=hi,d^c=dghi,e^c=egh,f^c=fghi,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, 46127);