#4386373

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1562 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)(9,10)(11,16)(12,13)(14,15)(19,20)(23,24)(25,26,29,32)(27,30,31,28);(2,5)(6,8)(10,15)(11,12)(18,22)(23,24)(25,27,29,31)(26,30,32,28);(3,7)(6,8)(9,11,14,12)(10,13,15,16)(17,18)(19,23)(20,24)(21,22)(25,28,29,30)(26,31,32,27);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,13,15,16)(17,19,21,20)(18,23,22,24)(25,26)(27,30)(28,31)(29,32);(1,3,4,7)(2,6,5,8)(9,13)(10,11)(12,15)(14,16)(17,20,21,19)(18,24,22,23)

Polycyclic presentation:

2^9|a^2=fg,b^2=fhi,c^2=g,d^2=h,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4293569);