#134633

order	512
nilpotency class	4
isoclinism	isoclinic class of #134475
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1804
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

294 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 100 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,c^2=gh,d^2=i,e^2=hi,f^2=ghi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fghi,g^c=gi,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, 41829);