#1786528

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2284 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1693724);