#706337

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2235 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=h,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, 613533);