#1248966

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1514 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=fg,d^2=fgh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1156162);