#561626

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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

2216 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:

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

Polycyclic presentation:

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

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, 468822);