#4375577

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

1629 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=fi,e^2=i,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, 4282773);