#4929348

order	512
nilpotency class	2
isoclinism	isoclinic class of #4928752
automorphism	Aut(G) order 2097152
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2360 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,14)(13,16)(19,23)(22,24)(25,26)(27,28)(29,32)(30,31);(9,10)(11,13)(12,14)(15,16)(17,18,21,20)(19,22,23,24)(26,31)(28,32);(1,2,4,6)(3,5,7,8)(9,10,12,14)(11,13,16,15)(17,19,21,23)(18,22,20,24)(25,27,30,29)(26,32,31,28);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,24)(21,23)(25,28,30,32)(26,27,31,29);(1,3,4,7)(2,5,6,8)(9,11,12,16)(10,13,14,15)(17,20,21,18)(19,24,23,22)(25,29,30,27)(26,32,31,28)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4836544);