#4432292

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

1658 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=fi,d^2=f,e^2=g,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, 4339488);