#488617

order	512
nilpotency class	4
isoclinism	isoclinic class of #488531
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3191	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#433 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

705 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,e^2=gi,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh

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, 395813);