#138924

order	512
nilpotency class	4
isoclinism	isoclinic class of #138844
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 ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

531 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 531 conjugacy classes of subgroups — too many to draw. Only its 80 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)(9,10,15,22,35,36,34,20,16,23,14,21,17,24,13,33)(11,25,28,18,32,19,26,39,29,40,27,38,30,37,12,31);(1,3,4,7)(2,5)(6,8)(9,11,17,32,16,29,35,30)(10,18,36,31,23,38,24,39)(12,13,28,14,26,34,27,15)(19,22,25,20,37,21,40,33);(9,12)(10,19)(11,13)(14,32)(15,30)(16,26)(17,28)(18,22)(20,31)(21,38)(23,37)(24,40)(25,36)(27,35)(29,34)(33,39)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=ghi,e^2=gh,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eghi,f^b=fh,h^b=hi,d^c=dghi,e^c=eghi,f^c=fghi,h^c=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46120);