#4428878

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1629 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgi,c^2=fh,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,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, 4336074);