#521390

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2554 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 428586);