#678220

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2214 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)(3,4)(5,6)(7,8)(9,10)(11,13)(12,15)(14,16)(19,23)(22,24)(25,26,31,30)(27,28,29,32);(2,6)(4,7)(9,11,14,12)(10,15,16,13)(17,18,21,20)(19,22,23,24)(25,27,31,29)(26,32,30,28);(17,19)(18,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32);(1,3)(2,7)(4,6)(5,8)(9,12,14,11)(10,15,16,13)(17,19)(18,22)(20,24)(21,23)(25,29,31,27)(26,32,30,28);(1,4,5,7)(2,3,6,8)(9,13,14,15)(10,11,16,12)(17,20,21,18)(19,24,23,22)(25,30,31,26)(27,32,29,28)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,d^2=fgi,e^2=fg,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, 585416);