#751600

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2210 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=fh,e^2=fhi,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, 658796);