#4131224

order	512
nilpotency class	2
isoclinism	isoclinic class of #4122169
automorphism	Aut(G) order 3145728
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

1746 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)(3,7)(4,6)(5,8)(9,10)(11,12)(13,14)(15,16)(17,18,20,23)(19,22,24,21)(26,30)(28,32);(1,3,4,8)(2,5,6,7)(9,10,13,14)(11,12,16,15)(17,19,20,24)(18,21,23,22)(26,30)(28,32);(2,6)(5,7)(10,14)(12,15)(17,18,20,23)(19,22,24,21)(25,26,29,30)(27,28,31,32);(1,3)(2,5)(4,8)(6,7)(9,11,13,16)(10,15,14,12)(17,19)(18,22)(20,24)(21,23)(25,27,29,31)(26,28,30,32);(1,2)(3,7)(4,6)(5,8)(9,12,13,15)(10,11,14,16)(17,18)(19,21)(20,23)(22,24)(25,28)(26,31)(27,30)(29,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=fg,d^2=fhi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4038420);