#4894594

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1806 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=fg,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=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, 4801790);