#1650997

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=i,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, 1558193);