#3966457

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2026 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=f,d^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3873653);