#4432230

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

2645 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)(3,5,8,7)(10,14)(11,12)(17,18,20,23)(19,21,24,22)(25,26,29,28)(27,31,32,30);(1,2)(3,5)(4,6)(7,8)(9,10)(11,16)(12,15)(13,14)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32);(9,11,13,12)(10,15,14,16)(17,19)(18,21)(20,24)(22,23)(25,27)(26,30)(28,31)(29,32);(1,2)(3,7)(4,6)(5,8)(9,12,13,11)(10,15,14,16)(17,19)(18,22)(20,24)(21,23)(25,28,29,26)(27,31,32,30);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,21)(20,23)(22,24)(25,27)(26,31)(28,30)(29,32)

Polycyclic presentation:

2^9|a^2=gi,c^2=fgi,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4339426);