#515488

order	512
nilpotency class	2
isoclinism	isoclinic class of #35346
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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

2590 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=f,e^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=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, 422684);