#624318

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2650 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=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, 531514);