#141722

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

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	C₂³	#493
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 278 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) = 80
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 80 points:

(1,2)(3,9)(4,12)(5,7)(6,11)(8,13)(10,15)(14,16)(17,18)(19,24,30,46)(20,33)(21,39)(22,26)(23,43)(25,36)(27,42)(28,40)(29,53,34,62)(31,59,57,73)(32,61,58,47)(35,52)(37,63)(38,72)(41,49)(44,66,48,55)(45,54,75,60)(50,70)(51,68)(56,80,67,77)(64,71)(65,69)(74,79,76,78);(1,3,4,11,5,10,13,16)(2,6,8,9,7,14,12,15)(17,19)(18,24,26,46)(20,34)(21,31)(22,30)(23,32)(25,48,50,44)(27,45,52,75)(28,47,43,61)(29,37)(33,62,63,53)(35,54)(36,66)(38,67)(39,59,49,73)(40,58)(41,57)(42,60)(51,76,64,74)(55,70)(56,69)(65,80,72,77)(68,79)(71,78);(2,7)(3,10)(8,12)(11,16)(17,20,35,68,23,38,21,36,22,37,42,71,40,69,41,70)(18,25,49,72,28,51,27,33,26,50,39,65,43,64,52,63)(19,29,54,78,32,56,31,55,30,34,60,79,58,67,57,66)(24,44,73,80,47,74,45,53,46,48,59,77,61,76,75,62)

Polycyclic presentation:

2^9|a^2=f,b^2=df,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, 48918);