#4310289

order	512
nilpotency class	2
isoclinism	isoclinic class of #4308740
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1778 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=gi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,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, 4217485);