#766884

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
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

1868 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fghi,c^2=fh,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 674080);