#141715

order	512
nilpotency class	4
isoclinism	isoclinic class of #141715
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1970
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

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

normal subgroups only

This group has 321 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)(11,16,22,37)(12,25)(13,31)(14,18)(15,34)(17,28)(19,27)(20,32)(21,43,26,51)(23,49,47,66)(24,50,48,38)(29,52)(30,62)(33,63)(35,56,39,45)(36,55,65,44)(40,61)(41,68)(42,59)(46,71,57,69)(53,58)(54,60)(64,70,67,72);(1,2,3,5,4,6,7,8)(9,11)(10,16,18,37)(12,26)(13,23)(14,22)(15,24)(17,39,40,35)(19,36,42,65)(20,38,34,50)(21,29)(25,51,52,43)(27,44)(28,56)(30,57)(31,49,63,66)(32,48)(33,47)(41,67,53,64)(45,61)(46,60)(54,71,62,69)(55,59)(58,72)(68,70);(2,6)(5,8)(9,12,27,58,15,30,33,61,14,29,59,68,32,60,13,28)(10,17,31,54,20,41,42,52,18,40,63,62,34,53,19,25)(11,21,44,70,24,46,47,56,22,26,55,72,48,57,23,45)(16,35,49,69,38,64,65,51,37,39,66,71,50,67,36,43)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=eh,d^2=gi,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg,d^b=di,f^b=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48911);