#134191

order	512
nilpotency class	4
isoclinism	isoclinic class of #133903
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2296 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

692 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,d^2=i,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ci,f^b=fgi,f^c=fhi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41387);