#553205

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

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

4598 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=fghi,d^2=fg,e^2=f,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, 460401);