#4416517

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2011 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:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=fi,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,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, 4323713);