#138929

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

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

normal subgroups only

This group has 542 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)(9,10,14,21,32,35,30,17,16,23,12,19,15,22,13,20)(11,24,27,38,33,36,26,37,29,39,25,18,28,34,31,40);(1,3,4,7)(2,5)(6,8)(10,17)(12,30)(13,14)(15,32)(18,34)(19,22)(20,23)(21,35)(24,37)(25,26)(27,31)(28,33)(36,38)(39,40);(9,11,16,29)(10,18,23,38)(12,31,14,26)(13,25,30,27)(15,33,32,28)(17,34,20,36)(19,39,21,24)(22,40,35,37)

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=i,d^2=ghi,e^2=gh,f^2=ghi,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, 46125);