#516446

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2526 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 423642);