#561495

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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

2686 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)(9,10,13,16)(11,12,15,14)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30);(2,4)(5,7,8,6)(9,11,13,15)(10,14,16,12)(19,25)(22,29)(26,31)(30,32);(1,2)(3,4)(9,12,13,14)(10,15,16,11)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32);(9,12)(10,11)(13,14)(15,16)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32);(9,11)(10,12)(13,15)(14,16)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=f,d^2=g,e^2=g,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, 468691);