#741836

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

1556 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)(9,10,14,15)(11,12,13,16)(27,31)(30,32);(10,15)(11,13)(18,22)(23,24)(25,26,29,28)(27,30,31,32);(1,2)(3,5)(4,6)(7,8)(9,11,14,13)(10,16,15,12)(17,18)(19,23)(20,24)(21,22)(25,27,29,31)(26,30,28,32);(1,2)(3,5)(4,6)(7,8)(9,12,14,16)(10,11,15,13)(17,19,21,20)(18,23,22,24)(25,26,29,28)(27,30,31,32);(1,3,4,7)(2,5,6,8)(9,13,14,11)(10,16,15,12)(17,20,21,19)(18,24,22,23)(25,28,29,26)(27,32,31,30)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fgi,d^2=ghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

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, 649032);