#678065

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=fg,d^2=ghi,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, 585261);