#503556

order	512
nilpotency class	4
isoclinism	isoclinic class of #501262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

758 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=efh,d^2=ei,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,g^c=gi

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, 410752);