#1521074

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2578 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(18,22)(19,23)(25,26)(27,30)(28,31)(29,32);(1,2,4,5)(3,7,8,6)(9,10,12,14)(11,13,15,16)(17,18)(19,20)(21,22)(23,24)(25,27)(26,29)(28,32)(30,31);(2,5)(6,7)(11,15)(13,16)(18,22)(19,23)(25,27,28,32)(26,30,31,29);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,13,14,16)(17,19,21,23)(18,20,22,24)(25,27)(26,30)(28,32)(29,31);(1,3,4,8)(2,6,5,7)(9,10)(11,13)(12,14)(15,16)(17,20)(18,23)(19,22)(21,24)(25,26)(27,30)(28,31)(29,32)

Polycyclic presentation:

2^9|b^2=fgh,c^2=ghi,d^2=fhi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1428270);