#505376

order	512
nilpotency class	4
isoclinism	isoclinic class of #501868
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×#469	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1753 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

509 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=fh,c^2=i,d^2=efh,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=dh,g^c=ghi,h^c=hi,g^d=gi

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, 412572);