#4483922

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
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

1718 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=i,d^2=g,e^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, 4391118);