#4456274

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
automorphism	Aut(G) order 1048576
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

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

Polycyclic presentation:

2^9|b^2=fhi,c^2=fi,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, 4363470);