#500338

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	#61261	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

734 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,18)(13,17)(14,32)(15,21)(19,29)(20,28)(22,39)(23,34)(24,36)(25,40)(26,33)(27,37)(30,35)(31,38);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,23)(13,30)(14,22)(15,27)(17,35)(18,34)(19,36)(20,33)(21,37)(24,29)(25,31)(26,28)(32,39)(38,40);(1,4,5,3)(2,6,7,8)(10,17)(11,22)(12,28)(13,29)(14,31)(16,33)(18,32)(19,20)(21,38)(23,25)(24,27)(26,30)(34,37)(36,39);(2,7)(6,8)(9,12,15,28)(10,18,21,20)(11,23,27,26)(13,29,31,14)(16,34,37,33)(17,19,38,32)(22,30,24,25)(35,36,40,39)

Polycyclic presentation:

2^9|b^2=e,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, 407534);