#561439

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

2302 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)(9,10,13,14)(11,17,18,15)(12,20,21,16)(19,22,24,23)(25,26)(27,32)(28,34)(29,33)(30,31)(35,39)(36,40)(37,38);(2,4)(5,7,8,6)(10,14)(11,18)(12,21)(22,23)(26,31)(27,34)(29,37)(36,39);(1,2)(3,4)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,27,30,34)(26,32,31,28)(29,35,37,40)(33,39,38,36);(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,28,30,32)(26,27,31,34)(29,36,37,39)(33,35,38,40);(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,29,30,37)(26,33,31,38)(27,35,34,40)(28,36,32,39)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fg,d^2=f,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, 468635);