#574937

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

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

3576 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=df,e^a=eg,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, 482133);