#500345

order	512
nilpotency class	4
isoclinism	isoclinic class of #499736
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	#61262	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

750 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)(10,16)(11,22)(12,28)(14,29)(15,30)(17,36)(19,31)(20,35)(23,37)(24,32)(25,33)(26,27)(34,38)(39,40);(1,3,5,4)(2,6,7,8)(9,10,13,16)(11,23,25,40)(12,18,28,21)(14,31,30,20)(15,19,29,35)(17,32,34,26)(22,39,33,37)(24,36,27,38);(1,4,5,3)(2,6,7,8)(9,11,13,25)(10,17,16,34)(12,24,28,27)(14,32,30,26)(15,33,29,22)(18,37,21,39)(19,23,35,40)(20,38,31,36);(2,7)(6,8)(9,12,13,28)(10,18,16,21)(11,24,25,27)(14,29,30,15)(17,37,34,39)(19,31,35,20)(22,26,33,32)(23,36,40,38)

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=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,h^b=hi,f^c=fh,g^c=ghi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407541);