#616551

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

2062 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 523747);