#558943

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
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

2162 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)(11,15)(12,16)(17,18,22,23)(19,24,25,20)(21,26,29,30)(27,31,32,28);(6,8)(9,10,13,14)(11,16,15,12)(18,23)(20,24)(21,29)(27,32);(1,2,4,3)(5,6,7,8)(9,11)(10,12)(13,15)(14,16)(17,19,22,25)(18,24,23,20)(21,27,29,32)(26,31,30,28);(1,3,4,2)(9,10)(11,12)(13,14)(15,16)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27);(9,12)(10,11)(13,16)(14,15)(17,21,22,29)(18,26,23,30)(19,27,25,32)(20,28,24,31)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,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, 466139);