#642156

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2102 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgh,d^2=fhi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,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, 549352);