#550837

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
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) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fg,d^2=fgi,e^2=gi,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,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, 458033);