#4456270

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
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

2219 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)(3,4)(5,7)(6,8)(10,14)(12,15)(17,18)(19,24)(20,22)(21,23)(27,31)(29,32);(1,3,6,5)(2,7,8,4)(9,10)(11,15)(12,16)(13,14)(18,20)(23,24)(25,26,28,30)(27,29,31,32);(2,8)(4,7)(10,14)(11,16)(17,19,22,21)(18,23,20,24)(25,26)(27,29)(28,30)(31,32);(1,4,6,7)(2,3,8,5)(9,11,13,16)(10,15,14,12)(17,20,22,18)(19,23,21,24);(1,5,6,3)(2,7,8,4)(9,12,13,15)(10,16,14,11)(17,21,22,19)(18,23,20,24)(25,27)(26,29)(28,31)(30,32)

Polycyclic presentation:

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

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, 4363466);