#679700

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

2014 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=f,d^2=fgi,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, 586896);