#141734

order	512
nilpotency class	4
isoclinism	isoclinic class of #141704
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)(12,27,31,17)(13,33)(14,36)(15,19)(16,24)(18,34)(20,29)(22,44)(23,45,52,71)(25,48)(26,56)(28,60,63,70)(30,65,66,67)(32,43,64,59)(35,37)(38,68,69,51)(39,49)(40,54)(41,62,57,72)(42,61,58,50)(46,55)(47,53);(9,11,22,49,14,26,13,24,15,25,34,55,29,53,35,54)(10,16,37,56,20,39,18,21,19,40,33,47,36,46,44,48)(12,28,61,68,32,52,66,72,31,63,50,51,64,23,30,62)(17,41,67,71,43,69,58,60,27,57,65,45,59,38,42,70);(1,2,6,8,4,7,3,5)(9,12,29,64,15,31,14,32)(10,17,20,43,19,27,36,59)(11,23,26,28,25,52,53,63)(13,30,22,50,35,66,34,61)(16,38,46,57,40,69,39,41)(18,42,33,67,44,58,37,65)(21,45,56,60,48,71,47,70)(24,51,55,62,54,68,49,72)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=egh,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, 48930);