#138930

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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 520 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)(9,10,15,21,16,22,14,33,17,23,34,40,32,39,13,20)(11,24,25,37,26,38,12,28,27,19,30,18,29,35,31,36);(1,3,4,7)(2,5)(6,8)(9,11,16,29,17,27,32,26)(10,18,22,28,23,37,39,36)(12,14,25,34,31,13,30,15)(19,20,38,21,24,33,35,40);(9,12)(10,19)(11,14)(13,27)(15,26)(16,25)(17,31)(18,20)(21,28)(22,38)(23,24)(29,34)(30,32)(33,37)(35,39)(36,40)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,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, 46126);