#4616411

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2340 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fhi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4523607);