#556768

order	512
nilpotency class	2
isoclinism	isoclinic class of #37436
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄²

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

3578 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=g,d^2=f,b^a=bf,c^a=cg,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, 463964);