#141708

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eg,d^2=g,e^2=hi,f^2=i,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, 48904);