#6433228

order	512
nilpotency class	3
isoclinism	isoclinic class of #29748
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
prime radical	C₂×#3050	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2048 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6340424);