#644115

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
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

1792 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=hi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,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, 551311);