#4423883

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1845 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=f,d^2=g,e^2=fg,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, 4331079);