#574943

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, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3350 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,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, 482139);