#6051943

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1781 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)(3,5,7,8)(10,11)(15,16)(18,21)(22,24)(27,31)(29,32);(3,7)(5,8)(9,10,14,11)(12,16,13,15)(19,23)(22,24)(25,26,28,30)(27,32,31,29);(1,3,4,7)(2,5,6,8)(9,11,14,10)(12,16,13,15)(17,18)(19,24)(20,21)(22,23)(25,26)(27,29)(28,30)(31,32);(9,12,14,13)(10,15,11,16)(17,19,20,23)(18,22,21,24)(25,27)(26,29)(28,31)(30,32);(1,2)(3,5)(4,6)(7,8)(9,13,14,12)(10,16,11,15)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=fhi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5959139);