#4474484

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
automorphism	Aut(G) order 1048576
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2442 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)(11,15,16,12)(17,18,20,22)(19,21,23,24)(25,26,30,27)(28,31,36,33)(29,34,37,32)(35,39,40,38);(1,2,4,6)(3,5,8,7)(11,16)(12,15)(19,23)(21,24)(26,27)(29,37)(31,33)(35,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,21)(20,23)(22,24)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,15)(14,16)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,3)(2,5)(4,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,21)(20,22)(23,24)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df

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, 4381680);