#515745

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

2574 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=f,d^2=fi,e^2=g,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, 422941);