#762851

order	512
nilpotency class	2
isoclinism	isoclinic class of #759288
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2146 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)(3,8,5,4)(9,10,14,12)(11,13,16,15)(17,18,20,22)(19,23,24,21)(26,30)(27,32);(1,3,6,5)(2,4,7,8)(9,11,14,16)(10,15,12,13)(17,19)(18,21)(20,24)(22,23);(2,7)(3,5)(11,16)(13,15)(17,18,20,22)(19,21,24,23)(25,26,29,30)(27,31,32,28);(1,4,6,8)(2,3,7,5)(9,12,14,10)(11,13,16,15)(17,19)(18,21)(20,24)(22,23)(25,27)(26,31)(28,30)(29,32);(1,5,6,3)(2,8,7,4)(9,13)(10,11)(12,16)(14,15)(17,18)(19,21)(20,22)(23,24)(25,28,29,31)(26,27,30,32)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,e^c=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, 670047);