#4341314

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
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

1516 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

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, 4248510);