#550845

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

2660 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)(17,18,23,25)(19,26,20,24)(21,22,31,32)(27,30,29,28);(6,8)(10,11)(14,15)(17,19,23,20)(18,24,25,26)(21,27,31,29)(22,28,32,30);(5,6)(7,8)(10,11)(14,15)(17,20,23,19)(18,24,25,26)(21,27,31,29)(22,30,32,28);(1,2,4,3)(9,11,13,10)(12,15,16,14)(17,21)(18,22)(19,27)(20,29)(23,31)(24,28)(25,32)(26,30);(1,3,4,2)(9,12,13,16)(10,14,11,15)(17,22)(18,21)(19,28)(20,30)(23,32)(24,27)(25,31)(26,29)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fg,d^2=fgi,e^2=fgi,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, 458041);