#719288

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

2438 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)(6,8)(9,10,13,16)(11,15,12,14)(17,18,22,21)(19,24,27,23)(20,28,30,25)(26,31,29,32);(6,8)(17,19,22,27)(18,23,21,24)(20,26,30,29)(25,31,28,32);(5,6,7,8)(9,11,13,12)(10,14,16,15)(17,19)(18,24)(20,29)(21,23)(22,27)(25,31)(26,30)(28,32);(1,2,3,4)(9,12,13,11)(10,15,16,14)(17,20,22,30)(18,25,21,28)(19,26,27,29)(23,31,24,32);(17,21,22,18)(19,23,27,24)(20,28,30,25)(26,31,29,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=g,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, 626484);