#1044973

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1772 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=fi,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 952169);