#612015

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

1734 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)(3,6)(4,5)(7,8)(9,10,13,12)(11,14,15,16)(17,18)(19,21)(20,22)(23,24)(25,26,28,29)(27,30,31,32);(9,11)(10,14)(12,16)(13,15)(17,19,20,23)(18,21,22,24)(26,29)(30,32);(2,5)(6,8)(11,15)(14,16)(25,27,28,31)(26,30,29,32);(9,12,13,10)(11,16,15,14)(18,22)(21,24)(26,29)(27,31);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=f,d^2=h,e^2=g,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, 519211);