#4416903

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
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

1642 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=fi,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, 4324099);