#141729

order	512
nilpotency class	4
isoclinism	isoclinic class of #141715
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×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	#33713	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 380 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=egh,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, 48925);