#4437058

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1450 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=g,d^2=hi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

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, 4344254);