#657498

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
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

2080 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)(9,10,13,16)(11,14,12,15)(17,18)(19,22)(20,21)(23,24)(26,28)(30,31);(1,3,6,4)(2,5,7,8)(9,11,13,12)(10,14,16,15)(18,21)(22,24)(25,26,29,28)(27,31,32,30);(9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24)(25,27)(26,30)(28,31)(29,32);(1,4,6,3)(2,5,7,8)(9,10)(11,14)(12,15)(13,16)(17,19,20,23)(18,22,21,24)(25,28,29,26)(27,31,32,30);(1,5)(2,3)(4,7)(6,8)(9,12,13,11)(10,15,16,14)(17,18)(19,22)(20,21)(23,24)(25,27)(26,30)(28,31)(29,32)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fhi,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=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, 564694);