#1650987

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1513 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fgi,d^2=fi,e^2=f,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, 1558183);