#4719654

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×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2025 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)(9,10,13,15)(11,12,16,14)(17,18,21,24)(19,20,23,22)(25,26,28,30)(27,29,31,32);(1,2,4,6)(3,5,8,7)(9,11)(10,14)(12,15)(13,16)(19,23)(20,22)(27,31)(29,32);(1,2)(3,5)(4,6)(7,8)(9,11,13,16)(10,14,15,12)(17,19,21,23)(18,22,24,20)(25,27,28,31)(26,29,30,32);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20,21,22)(18,23,24,19)(25,26)(27,29)(28,30)(31,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=fghi,c^2=fgi,d^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, 4626850);