#550717

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2672 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)(17,18,23,26)(19,24,20,29)(21,25,32,22)(27,28,30,31);(6,8)(10,11)(14,15)(17,19,23,20)(18,24,26,29)(21,30,32,27)(22,28,25,31);(5,6)(7,8)(10,11)(14,15)(17,20,23,19)(18,24,26,29)(21,30,32,27)(22,31,25,28);(1,2,4,3)(9,11,13,10)(12,15,16,14)(17,21)(18,25)(19,27)(20,30)(22,26)(23,32)(24,28)(29,31);(1,3,4,2)(9,12)(10,14)(11,15)(13,16)(17,22,23,25)(18,21,26,32)(19,28,20,31)(24,30,29,27)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fg,d^2=fgi,e^2=fi,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, 457913);