#556571

order	512
nilpotency class	2
isoclinism	isoclinic class of #37436
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

2254 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)(9,10,13,12)(11,14,15,16)(18,19)(23,25)(24,26)(30,31);(6,8)(10,12)(14,16)(17,18,22,19)(20,25,28,23)(21,26,29,24)(27,30,32,31);(5,6)(7,8)(11,15)(14,16)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31);(1,3,4,2)(9,11,13,15)(10,14,12,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31);(9,12,13,10)(11,16,15,14)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)

Polycyclic presentation:

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

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, 463767);