#732872

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2286 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=hi,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, 640068);