#507947

order	512
nilpotency class	4
isoclinism	isoclinic class of #499949
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

460 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,22)(12,25)(13,19)(14,29)(15,31)(16,21)(17,23)(18,27)(20,30)(24,37)(26,35)(28,34)(32,38)(33,40)(36,39);(1,3,4,8)(2,5,6,7)(9,11)(10,17,21,35)(12,16)(13,26)(14,23)(15,24)(18,34,37,20)(19,22,29,25)(27,33)(28,32)(30,39)(31,36,40,38);(9,12,16,11)(10,18,21,37)(13,26,14,23)(15,32,33,39)(17,34,35,20)(19,38,29,36)(22,31,25,40)(24,30,27,28);(2,6)(5,7)(9,13,16,14)(11,23,12,26)(15,30,33,28)(17,35)(20,34)(22,25)(24,32,27,39)(31,40)

Polycyclic presentation:

2^9|b^2=egh,c^2=i,d^2=g,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^c=ei,f^c=fh,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415143);