#556801

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

3642 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=f,d^2=fg,e^2=g,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, 463997);