#1293543

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2270 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fg,d^2=i,e^2=g,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, 1200739);