#141744

order	512
nilpotency class	4
isoclinism	isoclinic class of #141707
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1961
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#496
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

305 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 98 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)(4,6)(5,13)(7,12)(8,10)(11,15)(14,16)(17,18)(19,29,31,36)(20,33,24,25)(21,34)(22,40)(23,27)(26,38)(28,35)(30,44)(32,47,46,45)(37,43,42,48)(39,41);(2,6)(7,14)(8,13)(9,15)(17,19,30,43,35,46,39,25,23,31,38,48,22,32,21,33)(18,24,41,45,40,37,44,29,27,20,34,47,28,42,26,36);(1,3,10,16,4,11,5,12)(2,7,13,15,6,14,8,9)(17,20,35,42,23,24,22,37)(18,25,28,43,27,33,40,48)(19,26,32,34,31,44,46,41)(21,36,30,45,39,29,38,47)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=egh,d^2=g,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=cg,d^b=di,e^b=ei,f^b=fi,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48940);