#133977

order	512
nilpotency class	4
isoclinism	isoclinic class of #133889
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#537 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

737 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 737 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=di,e^b=ei,f^b=fg,d^c=di,f^c=fhi,f^d=fi,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, 41173);