#1386540

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2640 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fg,c^2=i,d^2=f,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=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, 1293736);