#1637731

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1812 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)(3,7)(4,5)(6,8)(9,10)(11,12)(13,14)(15,16)(18,21)(19,23)(25,26,30,31)(27,29,33,38)(28,32,37,39)(34,36,35,40);(2,5)(3,8)(9,10,13,14)(11,16,15,12)(17,18,20,21)(19,24,23,22)(26,31)(29,38)(32,39)(36,40);(2,5)(6,7)(10,14)(11,15)(17,18,20,21)(19,22,23,24)(25,27,30,33)(26,29,31,38)(28,35,37,34)(32,40,39,36);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,12,14,16)(17,18)(19,22)(20,21)(23,24)(25,28,30,37)(26,32,31,39)(27,34,33,35)(29,36,38,40);(1,3,4,8)(2,6,5,7)(9,12)(10,15)(11,14)(13,16)(17,19)(18,22)(20,23)(21,24)(25,29)(26,33)(27,31)(28,36)(30,38)(32,35)(34,39)(37,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=hi,c^2=fgi,d^2=fgh,e^2=ghi,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, 1544927);