#500148

order	512
nilpotency class	4
isoclinism	isoclinic class of #499763
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61051	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

980 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)(3,6)(4,5)(7,8)(9,10)(11,17)(12,19)(13,18)(14,30)(15,31)(16,21)(20,29)(22,39)(23,33)(24,35)(25,40)(26,37)(27,38)(28,34)(32,36);(2,5)(6,8)(9,11,16,26)(10,17,21,37)(12,23,15,32)(13,28,27,25)(14,22,29,24)(18,34,38,40)(19,33,31,36)(20,35,30,39);(1,3,4,7)(2,6,5,8)(10,18)(11,22)(12,15)(13,29)(14,27)(17,33)(19,20)(21,38)(23,28)(24,26)(25,32)(30,31)(35,39)(36,37);(2,5)(6,8)(9,12,16,15)(10,19,21,31)(11,23,26,32)(13,14,27,29)(17,33,37,36)(18,30,38,20)(22,25,24,28)(34,39,40,35)

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,d^c=di,f^c=fh,g^c=ghi,h^c=hi

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, 407344);