#3050828

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
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

2238 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)(11,16)(12,13)(14,15)(18,21)(19,23)(25,26,28,29)(27,30,31,32);(1,2,4,6)(3,5,8,7)(10,15)(13,16)(19,23)(22,24)(26,29)(27,31);(9,11,14,12)(10,16,15,13)(17,18)(19,22)(20,21)(23,24)(25,26)(27,30)(28,29)(31,32);(1,3,4,8)(2,5,6,7)(9,12,14,11)(10,16,15,13)(17,19)(18,22)(20,23)(21,24)(25,27,28,31)(26,30,29,32);(1,2)(3,7)(4,6)(5,8)(9,13,14,16)(10,12,15,11)(17,18)(19,22)(20,21)(23,24)(25,26)(27,32)(28,29)(30,31)

Polycyclic presentation:

2^9|a^2=h,b^2=fgi,c^2=ghi,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,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, 2958024);