#141735

order	512
nilpotency class	4
isoclinism	isoclinic class of #141707
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×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	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 374 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)(3,6)(7,8)(9,10)(11,21,25,43)(12,27,31,17)(13,33)(14,36)(15,19)(16,24,38,34)(18,22)(20,29)(23,45)(26,55,54,46)(28,59)(30,64,50,66)(32,42,63,58)(35,67)(37,68)(39,47,69,53)(40,61)(41,48,57,60)(44,49)(51,70)(52,72)(56,65)(62,71);(9,11,22,47,29,54,13,34,15,25,49,53,14,26,35,24)(10,16,33,55,36,69,18,43,19,38,67,46,20,39,44,21)(12,28,60,68,63,23,50,61,31,62,48,51,32,52,30,65)(17,40,64,72,58,37,57,71,27,56,66,45,42,70,41,59);(1,2,6,8,4,7,3,5)(9,12,29,63,15,31,14,32)(10,17,20,42,19,27,36,58)(11,23,26,28,25,52,54,62)(13,30,49,60,35,50,22,48)(16,37,69,56,38,70,39,40)(18,41,33,66,44,57,67,64)(21,45,55,59,43,72,46,71)(24,51,53,61,34,68,47,65)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,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, 48931);