#669473

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

2226 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)(3,5)(4,8)(6,7)(9,10,13,14)(11,12,15,16)(17,18,22,21)(19,24,23,20)(25,26,29,27)(28,30,32,31);(2,7)(5,8)(10,14)(11,15)(17,18,22,21)(19,20,23,24)(26,27)(28,32);(1,3,6,4)(2,5,7,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,24)(22,23)(25,27,29,26)(28,31,32,30);(1,4,6,3)(2,8,7,5)(9,12)(10,11)(13,16)(14,15)(17,20,22,24)(18,23,21,19);(1,5,6,8)(2,3,7,4)(9,11)(10,12)(13,15)(14,16)(17,21,22,18)(19,24,23,20)(25,28)(26,30)(27,31)(29,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=fgi,d^2=fgh,e^2=fgh,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, 576669);