#131702

order	512
nilpotency class	4
isoclinism	isoclinic class of #131668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#62489	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

518 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=hi,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fgh,h^b=hi,d^c=di,e^c=eg,f^c=fg

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38898);