#657521

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
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

1938 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 564717);