#579792

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2228 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 486988);