#4781475

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2433 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fgh,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4688671);