#133704

order	512
nilpotency class	4
isoclinism	isoclinic class of #133406
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂×#120	C₂×C₄
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

541 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 541 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(17,18)(19,27)(20,31,33,24)(21,34)(22,26)(23,29)(25,36)(28,42,44,48)(30,40)(32,39,47,41)(35,38)(37,43,45,46);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,22,30)(18,23,26,38)(20,28,33,44)(21,35,36,29)(24,37,31,45)(25,40,34,27)(32,46,47,43)(39,42,41,48);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,20,22,33)(18,24,26,31)(19,28,30,44)(21,32,36,47)(23,37,38,45)(25,41,34,39)(27,42,40,48)(29,43,35,46)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=i,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,d^b=di,e^b=ehi,f^b=fgi,d^c=dh,f^c=fi,e^d=ei

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, 40900);