#576049

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×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4640 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=i,d^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, 483245);