#141705

order	512
nilpotency class	4
isoclinism	isoclinic class of #141704
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

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

normal subgroups only

This group has 331 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(2,5)(3,7)(6,8)(9,10)(11,21)(12,27,30,17)(13,32)(14,36)(15,19)(16,24)(18,34)(20,33)(22,44)(23,45,53,71)(25,48)(26,56)(28,60,63,70)(29,64,66,67)(31,43,65,59)(35,37)(38,68,69,52)(39,49)(40,54)(41,62,57,72)(42,61,58,50)(46,55)(47,51);(9,11,22,49,14,26,13,24,15,25,34,55,33,51,35,54)(10,16,37,56,20,39,18,21,19,40,32,47,36,46,44,48)(12,28,61,68,31,53,66,72,30,63,50,52,65,23,29,62)(17,41,67,71,43,69,58,60,27,57,64,45,59,38,42,70);(1,2,3,5,4,6,7,8)(9,12,14,31,15,30,33,65)(10,17,36,59,19,27,20,43)(11,23,51,63,25,53,26,28)(13,29,34,61,35,66,22,50)(16,38,39,41,40,69,46,57)(18,42,37,64,44,58,32,67)(21,45,47,70,48,71,56,60)(24,52,49,72,54,68,55,62)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=eg,d^2=gi,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=cg,e^b=ei,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, 48901);