#678213

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
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

1744 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=f,d^2=fgi,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 585409);