#134347

order	512
nilpotency class	4
isoclinism	isoclinic class of #129430
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	#1799
prime radical	C₂⁴	#117
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 341 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) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=g,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,c^b=chi,f^c=fgi,f^e=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41543);