#611879

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1810 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,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, 519075);