#513560

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂²	C₂³×C₄²
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3134 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fg,d^2=g,e^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420756);