#719708

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

2462 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)(17,18,23,28)(19,25,20,24)(21,22,27,26)(29,30,31,32);(6,8)(17,19,23,20)(18,24,28,25)(21,29,27,31)(22,32,26,30);(5,6,7,8)(9,11,13,12)(10,14,16,15)(17,20,23,19)(18,25,28,24)(21,29,27,31)(22,32,26,30);(1,2,3,4)(9,12,13,11)(10,15,16,14)(17,21,23,27)(18,26,28,22)(19,29,20,31)(24,30,25,32);(17,22,23,26)(18,27,28,21)(19,30,20,32)(24,29,25,31)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=gh,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

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, 626904);