#553242

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3606 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=fg,e^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,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, 460438);