#146102

order	512
nilpotency class	4
isoclinism	isoclinic class of #146102
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2073
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

197 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 197 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

Polycyclic presentation:

2^9|a^2=eh,c^2=d,d^2=g,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fh,h^b=hi,e^c=eg,f^c=fgh,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53298);