#616149

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1965 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=gi,d^2=g,e^2=fh,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, 523345);