#5482641

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
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

1942 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fhi,c^2=fg,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=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, 5389837);