#854608

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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

1636 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fgi,c^2=fg,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 761804);